This document provides an overview of geometric construction concepts including:
- The principles of geometric construction using only a ruler and compass.
- Key terminology related to points, lines, angles, planes, circles, polygons and other basic geometric entities.
- Procedures for performing common geometric constructions such as bisecting lines, arcs and angles, constructing perpendiculars and parallels, dividing lines into equal parts, and constructing tangencies.
The document provides information for an engineering class including the instructor's name and class details, assignments due dates and details, and content on surveying techniques and geometric constructions. Key points covered include potential errors in surveying, definitions of surveying, examples of historical errors, instructions for groups to practice drawing techniques, and methods for drawing various geometric shapes and their intersections.
The document provides instructions for performing various geometric constructions using drawing instruments. It covers constructing lines, angles, triangles, quadrilaterals, circles, ellipses, parabolas, hyperbolas and their tangents. The methods include using a compass, set squares, concentric circles and the distance squared rule. Instructions are given step-by-step with diagrams to divide lines into ratios, bisect angles, construct perpendiculars, inscribe and circumscribe shapes, draw tangents and join two points with a curve. The document also introduces graphic language components, drawing instruments and their use in technical drawing and sketching.
This document provides instructions for performing various geometric constructions. It begins with an introduction on the importance of geometric constructions in engineering drawing. It then covers techniques for constructing lines, angles, triangles, circles, quadrilaterals, regular polygons, tangents to circles, joining circles, ellipses, and involutes. The document provides detailed step-by-step instructions for over 30 different geometric constructions, with diagrams to illustrate each method. Accuracy is emphasized as the main difficulty in geometric constructions.
This document provides instructions for performing various geometric constructions. It begins with introductory information on points, lines, and common geometric shapes. It then provides step-by-step instructions for constructing angles, triangles, circles, quadrilaterals, regular polygons, tangents to circles, joining circles, ellipses, involutes, and more. The constructions require only a compass and straightedge. Accuracy is emphasized as the key difficulty.
This document provides instructions for performing various geometric constructions involving points, lines, planes, circles, arcs, and polygons. It defines basic geometric elements like points and lines. It then describes how to construct lines and planes, as well as solids and curved surfaces. The remainder of the document outlines step-by-step processes for performing constructions like bisecting lines, drawing tangents, constructing regular polygons, inscribing and circumscribing shapes, and constructing intersections between curves.
The document provides an overview of Engineering Drawing - MEng 2031. It introduces basic concepts like graphical language, technical drawing, lettering, and line types. It describes standards for drawings and sheet sizes. It explains techniques for geometric constructions like triangles, polygons, circles, tangents, and ellipses. The purpose of engineering drawing is to communicate designs through accurate graphic representations.
Lecture4 Engineering Curves and Theory of projections.pptxKishorKumaar3
This document provides information about various types of plane curves generated by the motion of a circle or point rolling along another curve or line without slipping. It defines conic sections, which are curves formed by the intersection of a cone with a plane, including ellipses, parabolas, hyperbolas, and circles. It then discusses methods for constructing these conic sections geometrically using properties like eccentricity and foci. The document also covers roulettes like cycloids, epicycloids, hypocycloids, and trochoids formed by rolling motions, and provides examples of their geometric constructions.
This document provides an overview of topics related to engineering drawing and graphics. It covers scales, engineering curves, loci of points, orthographic projections, projections of points/lines/planes/solids, sections and developments, intersections of surfaces, and isometric projections. For each topic, it lists subsections that provide definitions, methods, and example problems. The document appears to be part of an online course or reference material for learning the principles and techniques of engineering drawing.
The document provides information for an engineering class including the instructor's name and class details, assignments due dates and details, and content on surveying techniques and geometric constructions. Key points covered include potential errors in surveying, definitions of surveying, examples of historical errors, instructions for groups to practice drawing techniques, and methods for drawing various geometric shapes and their intersections.
The document provides instructions for performing various geometric constructions using drawing instruments. It covers constructing lines, angles, triangles, quadrilaterals, circles, ellipses, parabolas, hyperbolas and their tangents. The methods include using a compass, set squares, concentric circles and the distance squared rule. Instructions are given step-by-step with diagrams to divide lines into ratios, bisect angles, construct perpendiculars, inscribe and circumscribe shapes, draw tangents and join two points with a curve. The document also introduces graphic language components, drawing instruments and their use in technical drawing and sketching.
This document provides instructions for performing various geometric constructions. It begins with an introduction on the importance of geometric constructions in engineering drawing. It then covers techniques for constructing lines, angles, triangles, circles, quadrilaterals, regular polygons, tangents to circles, joining circles, ellipses, and involutes. The document provides detailed step-by-step instructions for over 30 different geometric constructions, with diagrams to illustrate each method. Accuracy is emphasized as the main difficulty in geometric constructions.
This document provides instructions for performing various geometric constructions. It begins with introductory information on points, lines, and common geometric shapes. It then provides step-by-step instructions for constructing angles, triangles, circles, quadrilaterals, regular polygons, tangents to circles, joining circles, ellipses, involutes, and more. The constructions require only a compass and straightedge. Accuracy is emphasized as the key difficulty.
This document provides instructions for performing various geometric constructions involving points, lines, planes, circles, arcs, and polygons. It defines basic geometric elements like points and lines. It then describes how to construct lines and planes, as well as solids and curved surfaces. The remainder of the document outlines step-by-step processes for performing constructions like bisecting lines, drawing tangents, constructing regular polygons, inscribing and circumscribing shapes, and constructing intersections between curves.
The document provides an overview of Engineering Drawing - MEng 2031. It introduces basic concepts like graphical language, technical drawing, lettering, and line types. It describes standards for drawings and sheet sizes. It explains techniques for geometric constructions like triangles, polygons, circles, tangents, and ellipses. The purpose of engineering drawing is to communicate designs through accurate graphic representations.
Lecture4 Engineering Curves and Theory of projections.pptxKishorKumaar3
This document provides information about various types of plane curves generated by the motion of a circle or point rolling along another curve or line without slipping. It defines conic sections, which are curves formed by the intersection of a cone with a plane, including ellipses, parabolas, hyperbolas, and circles. It then discusses methods for constructing these conic sections geometrically using properties like eccentricity and foci. The document also covers roulettes like cycloids, epicycloids, hypocycloids, and trochoids formed by rolling motions, and provides examples of their geometric constructions.
This document provides an overview of topics related to engineering drawing and graphics. It covers scales, engineering curves, loci of points, orthographic projections, projections of points/lines/planes/solids, sections and developments, intersections of surfaces, and isometric projections. For each topic, it lists subsections that provide definitions, methods, and example problems. The document appears to be part of an online course or reference material for learning the principles and techniques of engineering drawing.
The document provides information on geometric constructions of conic sections. It defines conic sections as curves formed by the intersection of a plane and a cone, including circles, ellipses, parabolas, and hyperbolas. It then describes several methods for constructing each type of conic section geometrically, such as using concentric circles to draw an ellipse, the focus-directrix definition to draw parabolas, and locus properties involving distances from two focal points to draw hyperbolas. Diagrams illustrate each construction method.
The document discusses various geometric constructions including:
1. Dividing a line into equal parts and dividing a line in a given ratio.
2. Bisecting a given angle.
3. Inscribing a square in a given circle.
4. Drawing parabolas, cycloids, epicycloids, and hypocycloids by rolling and tracing circles.
Step-by-step methods are provided for each construction without mathematical proofs.
Conics Sections and its Applications.pptxKishorKumaar3
Conic sections are curves formed by the intersection of a cone with a plane. The type of conic section (triangle, circle, ellipse, parabola, hyperbola) depends on the position and orientation of the cutting plane relative to the cone's axis. Conic sections can be modeled and constructed using various methods that involve the focus, directrix, eccentricity, or arcs of circles. Roulettes are curves generated by the rolling contact of one curve on another without slipping, and include important types like cycloids, trochoids, and involutes that are used in engineering applications.
This document provides an overview of topics related to engineering graphics and geometric constructions. It covers scales, engineering curves, loci of points, orthographic projections, projections of geometric entities, sections and developments of solids, intersections of surfaces, and isometric projections. For each topic, it lists various subtopics and provides example problems and construction steps. The goal is to teach fundamental concepts and problem-solving techniques in engineering graphics.
Engg engg academia_commonsubjects_drawingunit-iKrishna Gali
1. The document discusses scales used in engineering drawings. It defines representative fraction and describes different types of scales including plane, diagonal, and triangular scales.
2. Construction techniques for various scales are provided, along with examples of how to construct a 1:4 scale and a diagonal scale of 3:200.
3. Common geometric constructions used in engineering drawings are also outlined, such as bisecting lines and angles, drawing perpendicular and parallel lines, and constructing regular polygons.
The document discusses two-dimensional loci and their construction. It defines locus as the path of a moving point or set of points satisfying given conditions. Examples of loci include circles, lines, ellipses, and arcs. The key types of loci are those where a point is a constant distance from a fixed point or line, equidistant between two fixed points, and bisecting the angle between two intersecting lines. Intersection of two loci occurs at points satisfying the conditions of both loci.
The document provides instructions for geometric constructions of various shapes and figures in engineering drawing, including:
- Lines, angles, arcs, polygons (regular shapes with equal sides like triangles, squares, pentagons, hexagons), conic sections, cycloidal curves, and involutes.
- Methods for constructing parallel lines, perpendicular lines, dividing a line into equal parts, bisecting angles, drawing arcs and circles through three points.
- Specific steps are outlined for constructing regular polygons like triangles, squares, pentagons, and hexagons given the length of their sides or the diameter of a circumscribing circle. The document also provides a method for constructing a regular polygon with any number of sides.
The document discusses various types of engineering curves including conics, cycloids, involutes, spirals, and helices. It provides definitions and construction methods for different conic sections obtained from intersecting a right circular cone with cutting planes in different positions relative to the axis. Specifically, it describes how triangles, circles, ellipses, parabolas, and hyperbolas can be obtained and provides step-by-step methods for constructing ellipses using concentric circles, rectangles, oblongs, arcs of circles, and rhombuses. It also demonstrates how to construct parabolas and use directrix-focus definitions to draw ellipses, parabolas, and other curves.
This document discusses basic geometry concepts including:
1) Points, lines, and planes are the building blocks of geometry. A point has position but no size, a line has no beginning or end, and a plane is a flat surface with length and breadth but no height.
2) Perpendicular lines meet at right angles and parallel lines are always the same distance apart and never meet.
3) The document provides exercises measuring and classifying angles, including perpendicular, parallel, and neither relationships between line segments as well as measuring and constructing various angles.
The document provides information about various plane curves covered in the Engineering Graphics course GE8152 Unit 1. It includes definitions and methods for constructing conic sections like circles, ellipses, parabolas and hyperbolas on a plane. It also describes construction of other curves like cycloids, epicycloids, hypocycloids, involutes of circles and squares. Detailed step-by-step processes are provided with diagrams to draw these curves using basic geometrical concepts like loci, tangents, normals and eccentricity.
The document provides information about various plane curves covered in the Engineering Graphics course GE8152 Unit 1. It includes definitions and methods for constructing conic sections like circles, ellipses, parabolas and hyperbolas on a plane. It also describes the construction of other curves like cycloids, epicycloids, hypocycloids, involutes of circles and squares. Detailed step-by-step processes are provided with diagrams to draw these curves using focus, directrix, radii and other geometric properties. The document serves as a reference for students to understand and construct different plane curves for applications in engineering graphics.
The document discusses several methods for drawing ellipses, including using a trammel, the parallelogram method, three-center and five-center arches. It also explains how to find the directions of tangents to an ellipse, including using an auxiliary circle or focal properties. Windows makes drawing ellipses easier than other conic sections like parabolas or hyperbolas.
The document discusses different types of engineering curves known as conics that are formed by the intersection of a cutting plane with a cone. These conics include ellipses, parabolas, hyperbolas, circles, and triangles. The document provides definitions, examples of uses, and methods for drawing each type of conic section. It specifically describes how to construct ellipses, parabolas, and hyperbolas using different geometric techniques such as the arc of a circle, concentric circles, directrix-focus, and rectangular methods.
Vaibhav Goel presented on circles and their properties. The presentation included definitions of key circle terms like radius, diameter, chord, and arc. It also proved several theorems: equal chords subtend equal angles at the center; a perpendicular from the center bisects a chord; there is one circle through three non-collinear points; equal chords are equidistant from the center; congruent arcs subtend equal angles; and the angle an arc subtends at the center is double that at any other point. The presentation concluded that angles in the same segment are equal and cyclic quadrilaterals have opposite angles summing to 180 degrees.
Hey to go back to m Al and resume to this email and resume y to the measure of interest in and out to go back and resume to this post to you y y y y y y y y y t y y y y y y to y y to y to the measure and resume for your help to learn to learn to learn to learn y y u yum install the world of SRP RQS PRQ PQS to go back to m Al to be a t shirt 👕 and go for the world 🌎 to be a t y to the to do the world i the world 🌎 and pink y to y y u can you must i and white 🤍 to be a I will be in and pink rod is Shreya aur Priyanshi and resume tow bar 🍺 to be a t y t y to the to go in I will be in the measure to be a t y t y to the world 🌍 to go to go with the world of interest in the morning 🌄🌄🌄🌄🌄🌄🌄🌅🌅🌅 to be a good mood to this email ✉️✉️📨 to go to go to go with you and your help in the morning 🌄 to be in and pink y to y y u yum 😋😋😋😋😋😋😋😋😋 to go to go to go to this email 📨📨📨 to go to be a good 😊😊😊 to go to go with you and pink and resume for your I I am in the measure to go to go I I I I I I I am in the measure of my friends go for it to go with you to get scholarship to this video 📸📸📸📸📸 to go to reel featuring David and resume for two 🕑🕑🕑 to go I have to check the to you y u sir to be in touch and white 🐻❄️🐻❄️ to be of any kind and white 🐻❄️🐻❄️ to go with you and your help and white 🐻❄️🐻❄️🐻❄️🐻❄️🐻❄️🐻❄️ to be in and out go back to you and pink y u can you send the measure of my to be a t to this video 📸📸📸 to go with you and pink y u sir to send the measure to be of interest in our country and white 🐻❄️🐻❄️🐻❄️🐻❄️ to go to you y y to y u yum 😋😋 to go to m go to sleep 😴😴😴😴😴 to be in touch with you must i to be of any other information to you must have been working in a little bit more time to get scholarship for two 🕑🕑🕑🕑 to be a little t to a t shirt 🎽🎽🎽🎽 to go to go to go to go to school 🏫🏫🏫 to be a little t y t to go to learn y u can be of SRP to be in the world 🌍🌍 to go to learn more about the measure to be of any action taken to be a little bit of interest for you must i to go with a t to go with the same shoes like to know about this video is currently a good 😊😊😊😊😊😊😊 to be of SRP to be in and out go talking with a t shirt 🎽🎽🎽 to go to go to go with a good 😊😊😊 to go to go to go to go with you must i to go with the measure of interest in the morning 🌅🌅 to go with a little more than a t y y y y y y y y y y y y y u sir to be of any action in and out go to sleep now and white striped shirt and they are in a t y y to go to go with the same to u dear and white striped shirt and pink y to go with a good 😊😊😊😊😊 to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go with the same to you must have been a little yyyyyyy to go to go to go to go to go to go to go to go to go to go to go to go to go to go to sleep 💤😴😴😴 and resume for your help to learn to learn by you and your family a very happy birthday dear fr
1. The document provides step-by-step instructions for constructing triangles using a compass and straightedge, including copying a triangle, constructing isosceles triangles given different parameters, constructing an equilateral triangle, and constructing medians and 30-60-90 triangles.
2. Key steps include marking points, using a compass to measure side lengths and draw arcs, and drawing lines between points to form the sides of triangles.
3. The instructions are illustrated with diagrams showing each step and the completed triangles.
The document discusses various methods of drawing conic sections such as ellipses, parabolas, and hyperbolas. It provides details on the concentric circle method, rectangle method, oblong method, arcs of circle method, and general locus method for drawing ellipses. For parabolas, it describes the rectangle method, tangent method, and basic locus method. The hyperbola can be drawn using the rectangular hyperbola method, basic locus method, and through a given point with its coordinates. The document also discusses how to draw tangents and normals to these conic section curves from a given point.
This document discusses various geometric constructions that can be performed using only a compass and ruler. It explains how to bisect angles and line segments, construct a 60 degree angle, and construct triangles given different combinations of side lengths or angles. Specifically, it provides step-by-step instructions on how to construct a triangle if given its base, one base angle, and the sum of the other two sides; or given its base, a base angle, and the difference between the other two sides; or given its perimeter and two base angles.
The document provides information on geometric constructions of conic sections. It defines conic sections as curves formed by the intersection of a plane and a cone, including circles, ellipses, parabolas, and hyperbolas. It then describes several methods for constructing each type of conic section geometrically, such as using concentric circles to draw an ellipse, the focus-directrix definition to draw parabolas, and locus properties involving distances from two focal points to draw hyperbolas. Diagrams illustrate each construction method.
The document discusses various geometric constructions including:
1. Dividing a line into equal parts and dividing a line in a given ratio.
2. Bisecting a given angle.
3. Inscribing a square in a given circle.
4. Drawing parabolas, cycloids, epicycloids, and hypocycloids by rolling and tracing circles.
Step-by-step methods are provided for each construction without mathematical proofs.
Conics Sections and its Applications.pptxKishorKumaar3
Conic sections are curves formed by the intersection of a cone with a plane. The type of conic section (triangle, circle, ellipse, parabola, hyperbola) depends on the position and orientation of the cutting plane relative to the cone's axis. Conic sections can be modeled and constructed using various methods that involve the focus, directrix, eccentricity, or arcs of circles. Roulettes are curves generated by the rolling contact of one curve on another without slipping, and include important types like cycloids, trochoids, and involutes that are used in engineering applications.
This document provides an overview of topics related to engineering graphics and geometric constructions. It covers scales, engineering curves, loci of points, orthographic projections, projections of geometric entities, sections and developments of solids, intersections of surfaces, and isometric projections. For each topic, it lists various subtopics and provides example problems and construction steps. The goal is to teach fundamental concepts and problem-solving techniques in engineering graphics.
Engg engg academia_commonsubjects_drawingunit-iKrishna Gali
1. The document discusses scales used in engineering drawings. It defines representative fraction and describes different types of scales including plane, diagonal, and triangular scales.
2. Construction techniques for various scales are provided, along with examples of how to construct a 1:4 scale and a diagonal scale of 3:200.
3. Common geometric constructions used in engineering drawings are also outlined, such as bisecting lines and angles, drawing perpendicular and parallel lines, and constructing regular polygons.
The document discusses two-dimensional loci and their construction. It defines locus as the path of a moving point or set of points satisfying given conditions. Examples of loci include circles, lines, ellipses, and arcs. The key types of loci are those where a point is a constant distance from a fixed point or line, equidistant between two fixed points, and bisecting the angle between two intersecting lines. Intersection of two loci occurs at points satisfying the conditions of both loci.
The document provides instructions for geometric constructions of various shapes and figures in engineering drawing, including:
- Lines, angles, arcs, polygons (regular shapes with equal sides like triangles, squares, pentagons, hexagons), conic sections, cycloidal curves, and involutes.
- Methods for constructing parallel lines, perpendicular lines, dividing a line into equal parts, bisecting angles, drawing arcs and circles through three points.
- Specific steps are outlined for constructing regular polygons like triangles, squares, pentagons, and hexagons given the length of their sides or the diameter of a circumscribing circle. The document also provides a method for constructing a regular polygon with any number of sides.
The document discusses various types of engineering curves including conics, cycloids, involutes, spirals, and helices. It provides definitions and construction methods for different conic sections obtained from intersecting a right circular cone with cutting planes in different positions relative to the axis. Specifically, it describes how triangles, circles, ellipses, parabolas, and hyperbolas can be obtained and provides step-by-step methods for constructing ellipses using concentric circles, rectangles, oblongs, arcs of circles, and rhombuses. It also demonstrates how to construct parabolas and use directrix-focus definitions to draw ellipses, parabolas, and other curves.
This document discusses basic geometry concepts including:
1) Points, lines, and planes are the building blocks of geometry. A point has position but no size, a line has no beginning or end, and a plane is a flat surface with length and breadth but no height.
2) Perpendicular lines meet at right angles and parallel lines are always the same distance apart and never meet.
3) The document provides exercises measuring and classifying angles, including perpendicular, parallel, and neither relationships between line segments as well as measuring and constructing various angles.
The document provides information about various plane curves covered in the Engineering Graphics course GE8152 Unit 1. It includes definitions and methods for constructing conic sections like circles, ellipses, parabolas and hyperbolas on a plane. It also describes construction of other curves like cycloids, epicycloids, hypocycloids, involutes of circles and squares. Detailed step-by-step processes are provided with diagrams to draw these curves using basic geometrical concepts like loci, tangents, normals and eccentricity.
The document provides information about various plane curves covered in the Engineering Graphics course GE8152 Unit 1. It includes definitions and methods for constructing conic sections like circles, ellipses, parabolas and hyperbolas on a plane. It also describes the construction of other curves like cycloids, epicycloids, hypocycloids, involutes of circles and squares. Detailed step-by-step processes are provided with diagrams to draw these curves using focus, directrix, radii and other geometric properties. The document serves as a reference for students to understand and construct different plane curves for applications in engineering graphics.
The document discusses several methods for drawing ellipses, including using a trammel, the parallelogram method, three-center and five-center arches. It also explains how to find the directions of tangents to an ellipse, including using an auxiliary circle or focal properties. Windows makes drawing ellipses easier than other conic sections like parabolas or hyperbolas.
The document discusses different types of engineering curves known as conics that are formed by the intersection of a cutting plane with a cone. These conics include ellipses, parabolas, hyperbolas, circles, and triangles. The document provides definitions, examples of uses, and methods for drawing each type of conic section. It specifically describes how to construct ellipses, parabolas, and hyperbolas using different geometric techniques such as the arc of a circle, concentric circles, directrix-focus, and rectangular methods.
Vaibhav Goel presented on circles and their properties. The presentation included definitions of key circle terms like radius, diameter, chord, and arc. It also proved several theorems: equal chords subtend equal angles at the center; a perpendicular from the center bisects a chord; there is one circle through three non-collinear points; equal chords are equidistant from the center; congruent arcs subtend equal angles; and the angle an arc subtends at the center is double that at any other point. The presentation concluded that angles in the same segment are equal and cyclic quadrilaterals have opposite angles summing to 180 degrees.
Hey to go back to m Al and resume to this email and resume y to the measure of interest in and out to go back and resume to this post to you y y y y y y y y y t y y y y y y to y y to y to the measure and resume for your help to learn to learn to learn to learn y y u yum install the world of SRP RQS PRQ PQS to go back to m Al to be a t shirt 👕 and go for the world 🌎 to be a t y to the to do the world i the world 🌎 and pink y to y y u can you must i and white 🤍 to be a I will be in and pink rod is Shreya aur Priyanshi and resume tow bar 🍺 to be a t y t y to the to go in I will be in the measure to be a t y t y to the world 🌍 to go to go with the world of interest in the morning 🌄🌄🌄🌄🌄🌄🌄🌅🌅🌅 to be a good mood to this email ✉️✉️📨 to go to go to go with you and your help in the morning 🌄 to be in and pink y to y y u yum 😋😋😋😋😋😋😋😋😋 to go to go to go to this email 📨📨📨 to go to be a good 😊😊😊 to go to go with you and pink and resume for your I I am in the measure to go to go I I I I I I I am in the measure of my friends go for it to go with you to get scholarship to this video 📸📸📸📸📸 to go to reel featuring David and resume for two 🕑🕑🕑 to go I have to check the to you y u sir to be in touch and white 🐻❄️🐻❄️ to be of any kind and white 🐻❄️🐻❄️ to go with you and your help and white 🐻❄️🐻❄️🐻❄️🐻❄️🐻❄️🐻❄️ to be in and out go back to you and pink y u can you send the measure of my to be a t to this video 📸📸📸 to go with you and pink y u sir to send the measure to be of interest in our country and white 🐻❄️🐻❄️🐻❄️🐻❄️ to go to you y y to y u yum 😋😋 to go to m go to sleep 😴😴😴😴😴 to be in touch with you must i to be of any other information to you must have been working in a little bit more time to get scholarship for two 🕑🕑🕑🕑 to be a little t to a t shirt 🎽🎽🎽🎽 to go to go to go to go to school 🏫🏫🏫 to be a little t y t to go to learn y u can be of SRP to be in the world 🌍🌍 to go to learn more about the measure to be of any action taken to be a little bit of interest for you must i to go with a t to go with the same shoes like to know about this video is currently a good 😊😊😊😊😊😊😊 to be of SRP to be in and out go talking with a t shirt 🎽🎽🎽 to go to go to go with a good 😊😊😊 to go to go to go to go with you must i to go with the measure of interest in the morning 🌅🌅 to go with a little more than a t y y y y y y y y y y y y y u sir to be of any action in and out go to sleep now and white striped shirt and they are in a t y y to go to go with the same to u dear and white striped shirt and pink y to go with a good 😊😊😊😊😊 to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go to go with the same to you must have been a little yyyyyyy to go to go to go to go to go to go to go to go to go to go to go to go to go to go to sleep 💤😴😴😴 and resume for your help to learn to learn by you and your family a very happy birthday dear fr
1. The document provides step-by-step instructions for constructing triangles using a compass and straightedge, including copying a triangle, constructing isosceles triangles given different parameters, constructing an equilateral triangle, and constructing medians and 30-60-90 triangles.
2. Key steps include marking points, using a compass to measure side lengths and draw arcs, and drawing lines between points to form the sides of triangles.
3. The instructions are illustrated with diagrams showing each step and the completed triangles.
The document discusses various methods of drawing conic sections such as ellipses, parabolas, and hyperbolas. It provides details on the concentric circle method, rectangle method, oblong method, arcs of circle method, and general locus method for drawing ellipses. For parabolas, it describes the rectangle method, tangent method, and basic locus method. The hyperbola can be drawn using the rectangular hyperbola method, basic locus method, and through a given point with its coordinates. The document also discusses how to draw tangents and normals to these conic section curves from a given point.
This document discusses various geometric constructions that can be performed using only a compass and ruler. It explains how to bisect angles and line segments, construct a 60 degree angle, and construct triangles given different combinations of side lengths or angles. Specifically, it provides step-by-step instructions on how to construct a triangle if given its base, one base angle, and the sum of the other two sides; or given its base, a base angle, and the difference between the other two sides; or given its perimeter and two base angles.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
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Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
1. Lecture 3: Geometric Construction
❑ Geometrical Constructions
❑ Construction Geometry
❑ The Ruler & Compass Geometry
FROM PAGE
CHAPTER 5
OF THE
PRESCRIBED
BOOK
4. ■ The accurate construction or drawing of basic drawing
primitives/entities (such as lines, angles, and shapes)
primarily USING A RULER AND A COMPASS ONLY
■ The relationship between the basic drawing primitives/entities
(e.g., lines, angles, and shapes)
■ Recognizing the geometry that exists within and between
objects to enable creation of solid models or multi-view
drawings
Geometric Construction is about:
5. Discussion Points
Points and Lines
Cartesian Coordinate System
Planes
Polygons
Basic geometric construction principles, e.g.,
bisection of Lines & Angles, etc.
Arcs & Tangency
6. TERMINOLOGIES #
POINTS
■ Points indicate exact locations in space and are represented
by a CROSS (and NOT A DOT) at the point of exact
location.
■ A point located on a line is represented by a short dash.
■ Points are considered dimensionless (that is, they have no
Height, Width or Depth)
■ Mathematically, points are defined on the Cartesian plane
by a set of (x, y) coordinates.
8. X
Y
0 1 2 3 4 5 6 7 8 9
3
2
1
0
6
5
4
A = X3, Y2
B = X4, Y4
C = X7, Y1
D = X8, Y5
A
B
D
C
SYSTEM
FOR POINTS
9. X
Y
0 1 2 3 4 5 6 7 8 9
3
2
1
0
6
5
4
A
B
D
C
A ref zero= X3, Y2
B ref A = X1, Y2
C ref B= X3, Y-3
D ref C= X1, Y4
COORDINATES SYSTEM
for POINTS
10. ■ LINE: the locus of a point between two or more
locations
■ A STRAIGHT LINE is the shortest distance between
two points.
■ Lines are considered to have length, but no other
dimension such as width or thickness.
■ Lines come in different forms
TERMINOLOGIES #
LINES
11. ■ Parallel lines
■ Two or more lines that are always the same distance
apart
■ Perpendicular lines
■ Two lines that are at a 90° angle
TERMINOLOGIES #
LINES
12. X
Y
0 1 2 3 4 5 6 7 8 9
3
2
1
0
6
5
4
A
D
B
C
Line AD
Line BC
SYSTEM
for LINES
13. X
Y
0 1 2 3 4 5 6 7 8 9
3
2
1
0
6
5
4
A
D
0°
90°
180°
270°
45°
❑ The polar system is relative system that references the previous
point and locates the next point by distance and an angle
Point “D” uses point “A” as the reference
point.
The polar coordinate of “D” is: 5.0 < 45°.
5.0 is its LENGTH,
45° is its ORIENTATION in the coordinate
plane.
for
LINES
15. ■ An ANGLE defines relative orientation of
two lines with respect to each other,
■ VERTEX
■ a Point at which two lines intersect
Vertex
TERMINOLOGIES #ANGLES
&VERTICES
19. #CONVERTING ANGLES FROM ONE
UNIT TO ANOTHER
To flip 48.30750 back to degrees, minutes, and
seconds
▪ 48.30750 = 480 + (0.3075 x 60 = 18.45)Ꞌ
▪ 480 (18.45)Ꞌ = 480 18Ꞌ + (0.45 x 60)ꞋꞋ
▪ 480 18Ꞌ 27ꞋꞋ
20. ■ Planes are defined by:
▪Three points not lying in a straight line
▪Two parallel lines
▪Two intersecting lines
▪A point and a line
4. BASIC GEOMETRIC ENTITIES
#PLANES
21. Planes are defined by the axes that lie on the plane.
Y
Z X
The XZ Plane
X
Y
Z
The XY Plane.
Y
X
Z
The YZ Plane
# CARTESIAN COORDINATES SYSTEM
#of
PLANES
22. ■ Radius
■ Distance from the
center of a circle to its
edge
■ Diameter
■ Distance across a circle
through its center
■ Circumference
■ Distance around the
edge of a circle
■ Chord
■ Line across a circle
that does not pass at
the circle’s center
5. BASIC GEOMETRIC ENTITIES
#CIRCLES
■ Has 360°
■ Quadrant
■ One fourth (quarter)
of a circle
■ Measures 90°
■ Concentric
■ Two or more circles
of different sizes that
share the same
center point
23. Defined as a closed plane figure with three or
more straight lines:
❑ Triangles
❑ Quadrilaterals
❑ Hexagons
❑ Octagons
6. BASIC GEOMETRIC ENTITIES
#POLYGON
24. BASIC GEOMETRIC ENTITIES #BISECTING
LINES
1. Given a line AB
1. With the centre A and a radius
greater than half AB, draw arcs
on both sides of AB
1. With the centre B and the same
radius, draw arcs intersecting
the previous arcs at C and D
1. Draw a line joining C and D and
cutting AB at E
1. AE = EB = 1/2*AB
1. Note that CD bisects AB at a
right angle
A B
C
D
E
900
25. BASIC GEOMETRIC ENTITIES #BISECTING
ARCS
1. Given an arc AB
1. With the centre A and a radius
greater than half AB, draw arcs
on both sides of AB
1. With the centre B and the same
radius, draw arcs intersecting
the previous arcs at C and D
1. Draw a line joining C and D and
cutting AB at E
1. AE = EB = 1/2*AB
1. Note that CD passes through
centre O of arc
A B
C
D
E
O
26. B.G.E. #BISECTING ANGLES
1. Given two lines AB and BC
1. With B as centre and a
convenient radius, draw an arc
cutting AB at D and BC at E
1. With the centres D and E and
the same or any convenient
radius, draw arcs intersecting F
1. Draw a line joining B and F,
1. BF bisects the angle BAC, i.e.
ABF = FBC
A
C
B
D
E
F
27. B.G.E. #PERPENDICULARS TO GIVEN LINES
1. Given a line AB and a point P
near the MIDDLE of the line
1. With the P as centre and any
convenient radius R1 draw an
arc cutting AB at C and D
1. With radius R2 greater than R1
draw arcs intersecting each
other at O
1. Draw a line joining O and P and
cutting AB at E
1. Then PO is the required
perpendicular
A B
C D
P
900
O
R1 R1
R2 R2
28. B.G.E. #PERPENDICULARS TO GIVEN LINES
1. Given a line AB and a point P
near the END of the line
1. With a point O selected in
space, draw an arc with a
radius equal OP, larger than a
semicircle, cutting AB at C
1. Draw a line joining C and O, and
produce it to cut the arc at Q
1. Draw a line joining P and Q
1. Then PQ is the required
perpendicular
O
B
C P
900
Q
A
29. B.G.E. #PERPENDICULARS TO GIVEN LINES
1. Given a line AB and a
point P outside AB
2. With centre A and a
radius AP, draw an arc
EF, cutting AB or an
extension of AB at C
1. With centre C and a
radius equal to CP,
draw an arc cutting EF,
at D
1. draw a line joining D
and P, intersecting AB
at Q
1. Then PQ is the required
perpendicular
P
B
C
900
Q
A
F
E
D
30. B.G.E. # PARALLEL LINES GIVEN A POINT
1. Given a line AB and a
point P outside AB
2. With centre P and a
convenient radius R1,
draw an arc CD cutting
AB at E
1. With centre E and
same radius equal to
EP, draw an arc cutting
AB, at F
2. With centre E and
same radius equal to
FP, draw an arc cutting
CD, at Q
1. draw a straight line
joining P and Q, which
is required parallel line
P
B
C
Q
A F
E
D
R1
31. B.G.E. # DIVIDING LINES
1. Given a line AB draw another line
AC making an angle of less than 30
with AB
1. With the help of a compass, mark
7 equal parts of an suitable length
on line AC and mark them by
points 1’, 2’, 3’, 4’, 5’, 6’ and 7’.
1. Join the last point 7’ with point B
of the line AB
2. Now from each of the other
marked points 6’, 5’, 4’, 3’, 2’, and
1’, draw lines parallel to 7’B
cutting the line AB at 6, 5, 4, 3, 2,
and 1 respectively
3. Now the line AB has been divided
into 7 equal parts. You can verify
this by measuring the lengths
B
A
1 2
3
4 5
6 7
1’
2’
3’
4’
5’
6’
7’
C
32. B.G.E. # BISECT ANGLES
1. Given an angle ABC, with B as
centre and any radius draw an arc
cutting AC at D and BC at E
1. With D and E as centres and the
same or any convenient radius,
draw arcs intersecting each other
at F.
1. Join B and F. BF bisects the angle
ABC, i.e., ABF = FBC
C
B E
D F
A
33. B.G.E. # TRISECT A GIVEN RIGHT ANGLE
1. Given the right angle ABC, and any
radius, draw an arc cutting AB at D
and BC at E
1. With the same radius and D and E
as centres, draw arcs cutting the
arc DE at points Q and P.
2. Draw lines joining B with P and Q.
BP and BQ trisect the right angle
ABC. ABP = PBQ = QBC = 1/3 ABC
C
B E
D
A
P
Q
34. B.G.E. # DRAW A LINE AT ANGLE TO ANOTHER
1. Given PQ, and an angle AOB, with
centre O and any radius, draw an
arc cutting OA at C and OB at D
1. With the same radius and P as
centre, draw an arc EF cutting PQ
at F.
2. With F as centre and radius equal
to CD, draw an arc cutting the arc
EF at G.
1. from P draw a line passing through
G, to obtain the required line
B
O
D
C
A
Q
P F
G
E
35. B.G.E. # FIND THE CENTRE OF AN ARC
1. Given the arc AB, draw two chords
A and B of any length.
1. Draw perpendicular bisectors of
CD and EF intersecting each other
at O, then O is the required centre
B
O
D
A
C E
F
36. B.G.E. # DRAW AN ARC TOUCHING 2 STRAIGHT LINES GIVEN
THE RADIUS
1. Given the lines AB and AC, and a
radius, R, use the radius and the
centre A to draw two arcs cutting
AB at P and AC at Q.
1. With P and Q as centres and the
same radius, draw arcs
intersecting each other at O
1. With O as centre and radius equal
to R, draw the required arc.
B
P O
A
C
R
R
Q
R
38. To draw the arc, we must find the location of the center of that arc.
How do we find the center of the arc?
B.G.E. #FILLETS AND ROUNDS
39. To draw an arc of given radius tangent to two perpendicular lines
Given arc radius r
r
r
B.G.E. #FILLETS AND ROUNDS
40. center of the arc
Starting point
Ending point
To draw an arc of given radius tangent to two perpendicular lines
Given arc radius r
B.G.E. #FILLETS AND ROUNDS
41. +
+
r
r
To draw an arc of given radius tangent to two perpendicular lines
Given arc radius r
B.G.E. #FILLETS AND ROUNDS
42. T.P.1
T.P.2
To draw an arc of given radius tangent to two lines
Given arc radius r
B.G.E. #ARCS AND TANGENCIES
43. C
To draw a line tangent to a circle at a point on the circle
Given
B.G.E. #ARCS AND TANGENCIES
44. C
mark a tangent point
To draw a line tangent to a circle from a point outside the circle
Given
B.G.E. #ARCS AND TANGENCIES
45. C1
C2
Tangent point
R1
R2
The center of two circles and tangent point must lie on the same
straight line !!!
A circle tangent to another circle
B.G.E. #ARCS AND TANGENCIES
46. To draw a circle tangent to two circles I
+
C
2
Given
+
C
1
C
+
Example
47. +
+
C
1
C2
R +
R1
Given
Two circles and the radius of the third circle = R
R +
R2
R1
R
2
C
center of the arc
To draw a circle tangent to two circles I
R
48. C2
R2
When circle tangent to other circle
C1
Tangent point
R1
The center of two circles and tangent point must lie on the
same straight line !!!
50. + +
C
1
C
2
R –
R2
To draw a circle tangent to two circles II
Given
Two circles and the radius of the third circle = R
R –
R1
R
1
R
2
C
R
51. +
C
1
+ C
2
To draw a circle tangent to two circles III
Given
Two circles and the radius of the third circle = R
R +
R2
R –
R1
R
1
R
2
C
52. Polygons can be:
#DRAWING POLYGONS
■ inscribed (drawn within a
circumference) or
■ circumscribed (drawn
around a circumference)
53. An inscribed polygon is
constructed for polygons
where the number of sides
and the distance across the
corners have been given.
6-sided polygon
Draw a 6-sided polygon, of a length x across corners.
• Draw a line AB equal to the given length, x
• With the centre A and radius AB draw a semi-circle BP
• Divide the semicircle into 6 equal parts ( 180° ÷ 6 = 30°
) which can be easily done by using the triangles (in this
case the 30° - 60° triangle). Number the divisions as 1, 2,
etc starting at P
• Draw a line joining A with point 2
• Draw a perpendicular bisector of line A2 and another
perpendicular bisector of line AB, and mark their
intersection as O (however in this particular example that
intersection is at point 4
• With centre O/4 and radius OA/4A, draw a circle
• With radius AB and starting from B, cut the circle at
points C, D,….2
• Draw lines BC, CD, etc thus completing the required
polygon
#INSCRIBED POLYGON
C
D
E
55. A circumscribed polygon can
be constructed by determining
the number of sides and the
distance across the flats.
6-sided polygon
Draw a 6-sided polygon, of a length x across flats/sides.
• Draw a horizontal line AB equal to the given length, x,
and bisect it and mark the mid point O
• With the centre O and radius OA or OB draw a circle
• Divide the circle into 6 equal parts ( 360° ÷ 6 = 60° )
which can be easily done by using the triangles (in this
case the 30° - 60° triangle) to locate points C, D, E, & F.
• Draw tangents at points A, B, C, D, E, & F to intersect at
points 1, 2, 3, ….6.
• Draw a perpendicular bisector of line A2 and another
perpendicular bisector of line AB, and mark their
intersection as O (however in this particular example that
intersection is at point 4
• With centre O/4 and radius OA/4A, draw a circle
• With radius AB and starting from B, cut the circle at
points C, D,….2
• Draw lines BC, CD, etc thus completing the required
polygon
#CIRCUMSCRIBED POLYGON
58. ■ Equilateral
■ All three sides are of equal length and all three angles are
equal
■ Isosceles
■ Two sides are of equal length
■ Scalene
■ Sides of three different lengths and angles with three
different values
■ Right Triangle
■ One of the angles equals 90°
■ Hypotenuse: side of a right triangle that is opposite the 90°
angle
6. BASIC GEOMETRIC ENTITIES
#TRIANGLES
59. Quadrilaterals are figures with fours sides
■ Square
■ Four equal sides and all angles equal 90°
■ Rectangle
■ Two sides equal lengths and all angles equal 90°
■ Trapezoid
■ Only two sides are of equal length
■ Rhombus
■ All sides are equal length and opposite angles are equal
■ Rhomboid
■ Opposite sides are equal length and opposite angles are equal
■ Trapezium – No sides parallel
7. BASIC GEOMETRIC ENTITIES
#QUADRILATERALS
60. Construction Principles
• Bisecting a Line
1. Draw an arc about two-thirds of the line
length from each end.
2. Locate and label the two points where the
arcs intersect.
3. Draw a line between the two labeled points.
This line now bisects the original line.
61. Construction Principles
• Bisecting an Angle
1. Draw an arc at a convenient length from the
angle vertex.
2. Locate and label the two points where the
arc intersects the angle lines.
3. Draw the same size arc from each
intersection. At the intersection of the new
arcs, draw a line to the angle vertex.
62. Construction Principles
• Drawing an Arc or Circle with Three Points
1. Draw a line from the middle point to each of
the other points.
2. Bisect each of the lines.
3. Where the bisect lines intersect is the center
of the circle.
4. Draw the circle using the new center point.
63. Construction Principles
• Drawing Parallel Lines
1. Set the compass to the required parallel
distance.
2. Draw an arc from each end of the given line
on the same side of the line.
3. Connect the top of the arcs with a line. This
will be parallel to the given line.
64. Construction Principles
• Drawing Perpendicular Lines
1. Draw an arc that crosses the given line in
two places from a given point on the line.
2. Draw a larger arc from each point where
the first arc intersects the given line.
3. Where the new arcs intersect, draw a
line to the original point on the line.
65. Construction Principles
• Equally Dividing a Line
1. Draw a perpendicular line from one end of the
given line.
2. From the other end of the given line, measure a
distance that can be easily divided to the
perpendicular line.
3. Put tic marks on the angled line at the division
increments.
4. Transfer these marks perpendicularly to the given
line.
66. Construction Principles
• Transferring Shapes
1. Label each point on the original shape.
2. Select a start point, move clockwise.
3. Set the compass to the distance from the start
point to the second point and lightly draw an arc at
the new location.
4. Proceed in the same manner to the other points on
the shape.
5. Connect all the points with a line once they have
been identified.
67. Polygon Construction Principles
• Drawing a Square
1. Lightly draw center lines of a circle.
2. Lightly draw a circle with the diameter as the size of
the square.
3. Draw construction lines parallel to the center lines
and tangent to the circle.
4. If the four sides are equal, darken in the lines.
Square
68. • Drawing a Pentagon (five sides)
1. Lightly draw a circle with the diameter as the size
of the pentagon.
2. From the top of the circle, draw a 72° line from the
vertical axis.
3. Where this line intersects the circle, draw a
horizontal line. Set the compass for the length of
the line in the circle.
4. Transfer the compass distance around the circle
and connect the points.
Penta
gon
Polygon Construction Principles
69. • Drawing a Hexagon (six sides)
1. Lightly draw a circle with the diameter as the size
of the hexagon.
2. Draw horizontal lines tangent to the top and
bottom of the circle.
3. Draw 60° lines from the horizontal lines, so they
are tangent to the circle.
4. Darken all lines. Hexag
on
Polygon Construction Principles
70. Polygon Construction Principles
• Drawing an Octagon (eight sides)
1. Lightly draw a circle with the diameter as the size
of the octagon.
2. Draw two horizontal and two vertical lines tangent
to the circle.
3. Draw 45° lines so they are tangent to the circle.
4. Darken all lines.
Octa
gon
71. Constructing Tangent Arcs
• Arc within an Angle
1. Set the compass to the given radius.
2. Swing an arc from each line in the angle.
3. Draw tangent parallel lines from the given
angle lines to the arcs.
4. At the intersection of the lines, draw the
arc within the angle.
72. Constructing Tangent Arcs
• Arc between a Curve and a Line
1. Set the compass to the given radius.
2. Swing an arc from the line and draw a
parallel line.
3. Swing an arc from the outside of the circle
and draw a concentric circle through the arc.
4. At the intersection of the line and the arc,
draw the new arc between the curve and the
line. Use the given radius.
73. Constructing Tangent Arcs
• Arc between Two Curves
1. Set the compass to the given radius.
2. For each curve, swing an arc from the
outside of each curve and draw a concentric
arc through the first arc.
3. At the intersection of the arcs, draw a new
arc between the curves. Use the given
radius.
74. Conic Sections
• A conic section is formed by passing a plane
through a cone.
Triangle: formed when plane passes vertically
through the apex of a cone
Circle: formed when plane passes horizontally
through a cone
Ellipse: formed when plane passes through at an
incline
75. Conic Sections
• A conic section is formed by passing a plane
through a cone.
Parabola: formed when plane passes vertically
through a cone against its axis
Hyperbola: formed when plane passes vertically
through a cone
76. Conic Sections
• References to Drawing Irregular Shapes
(see Chapter 4)
– Drawing an ellipse
• Concentric circle method
• Trammel ellipse
• Foci ellipse
– Drawing a parabola
– Drawing a hyperbola
77. Conic Sections
• References to Drawing Irregular Shapes
(see Chapter 4)
– Drawing a spiral
– Drawing a helix
– Drawing an involute
– Drawing a cycloid curve