En esté trabajo vamos a ver que las transformaciones se pueden considerar como sistemas de coordenadas definidos respecto a un sistema de coordenadas global, y veremos la ventaja de comprender esta dualidad en el ámbito de los gráficos por ordenador
The document discusses different types of coordinate systems, including Cartesian and polar coordinates. It provides definitions and examples of transforming coordinates between the Cartesian and polar systems. The key steps for transformation involve using trigonometric functions like sine, cosine, and inverse tangent. Translating and rotating coordinate axes is also covered, with equations provided for finding new coordinates after such transformations. Examples of graphs in polar coordinates include a parabola and circle.
The document discusses various topics relating to the transformation of coordinates including:
1) The transformation of coordinates is a process of changing a relationship, expression, or figure into another following a given law, which is analytically expressed through one or more transformation equations.
2) It describes how to convert between rectangular and polar coordinates using trigonometric functions.
3) It also explains how to perform translations and rotations of coordinate axes, which are important transformations.
Un sistema de coordenadas es un conjunto de valores que permiten definir la posición de cualquier punto de un espacio vectorial.
Estos sistemas de coordenadas son de suma importancia ya que para resolver problemas de electrotástica, magnetostática y
campos variables en el tiempo, tenemos que tener un conocimiento previo de cómo utilizarlos y cómo hacer cambios de bases
vectoriales entre ellos para que la resolución de los problemas sea menos compleja.
TRANSFORMACIÓN DE COORDENADAS
Para adentrarnos en el tema de transformación de coordenadas, considero que es importante conocer primeramente, la definición y/u concepto
de lo que es un sistema de coordenadas, así que iniciando desde este punto, tenemos que:
Un sistema de coordenadas es un sistema que utiliza uno o más números (coordenadas) para determinar unívocamente la posición de un
punto u objeto geométrico.
El orden en que se escriben las coordenadas es significativo y a veces se las identifica por su posición en una tupla ordenada; también se las
puede representar con letras, como por ejemplo (la coordenada-x). El estudio de los sistemas de coordenadas es objeto de la geometría
analítica, permite formular los problemas geométricos de forma "numérica“.
Teniendo esto en claro, podemos definir a aquello que se conoce como Transformación de coordenadas… Entonces, tenemos que:
La transformación de coordenadas es una operación por la cual una relación, expresión o figura se cambia en otra siguiendo una ley dada.
Analíticamente, la ley se expresa por una o mas ecuaciones llamadas ecuaciones de transformación.
También se define como el cambio de posición de los ejes de referencia en un sistema de coordenadas, ya sea por traslación, rotación, o ambas. El propósito de dicho cambio por lo general es simplificar la ecuación de una curva para manejo posterior.
TRANSFORMACIÓN DE COORDENADAS RECTANGULARES A POLARES
Primero definiremos a cada sistema de coordenadas…
Coordenadas Rectangulares:son aquellas que nos permiten determinar la ubicación de un punto mediante dos distancias y refiriéndolas a una dirección base y a un punto base.
The document discusses coordinate transformations between rectangular and polar coordinate systems. It provides the key formulas for converting between the two systems using trigonometric relationships. Specifically, it gives the formulas to transform from rectangular to polar coordinates as x=rcosθ and y=rsinθ, and from polar to rectangular as x=rcosθ and y=rsinθ, where r is the distance from the origin and θ is the angle relative to the x-axis. It also discusses the process of translating and rotating coordinate axes.
Erika vidal 20% transformacion de coordenadaserikavidal14
This document discusses transformations of coordinate systems, including translation and rotation of axes. It provides examples of converting between rectangular and polar coordinates. Specifically:
1) It explains how to convert between rectangular (x,y) and polar (r,θ) coordinates using trigonometry.
2) Examples are given of converting coordinate points between rectangular and polar formats.
3) Translation of axes is defined as changing the reference axes without rotation, keeping each axis parallel to its original position.
4) Rotation of axes changes the orientation of the coordinate plane while preserving the geometric shape of curves.
The document discusses coordinate transformations, including:
- Converting between rectangular (x,y) and polar (r,θ) coordinate systems using trigonometry and the Pythagorean theorem.
- Translating coordinate axes by shifting the origin point without rotating the axes.
- Rotating coordinate axes by fixing the origin and rotating the x and y axes by an angle using trigonometric formulas.
- Examples are provided of transforming points between rectangular and polar coordinates and translating and rotating coordinate axes. The purpose is to simplify equations for further analysis.
The document introduces concepts of vector algebra and geometry in two dimensions. It defines vectors, vector operations like addition and scalar multiplication, and properties including equality of vectors, parallelism, orthogonality, and the Cauchy-Schwarz inequality. It also covers vector representations geometrically as arrows, vector length, unit vectors, and linear combinations of vectors. Key concepts are illustrated with diagrams of vectors in the Cartesian plane.
The document discusses different types of coordinate systems, including Cartesian and polar coordinates. It provides definitions and examples of transforming coordinates between the Cartesian and polar systems. The key steps for transformation involve using trigonometric functions like sine, cosine, and inverse tangent. Translating and rotating coordinate axes is also covered, with equations provided for finding new coordinates after such transformations. Examples of graphs in polar coordinates include a parabola and circle.
The document discusses various topics relating to the transformation of coordinates including:
1) The transformation of coordinates is a process of changing a relationship, expression, or figure into another following a given law, which is analytically expressed through one or more transformation equations.
2) It describes how to convert between rectangular and polar coordinates using trigonometric functions.
3) It also explains how to perform translations and rotations of coordinate axes, which are important transformations.
Un sistema de coordenadas es un conjunto de valores que permiten definir la posición de cualquier punto de un espacio vectorial.
Estos sistemas de coordenadas son de suma importancia ya que para resolver problemas de electrotástica, magnetostática y
campos variables en el tiempo, tenemos que tener un conocimiento previo de cómo utilizarlos y cómo hacer cambios de bases
vectoriales entre ellos para que la resolución de los problemas sea menos compleja.
TRANSFORMACIÓN DE COORDENADAS
Para adentrarnos en el tema de transformación de coordenadas, considero que es importante conocer primeramente, la definición y/u concepto
de lo que es un sistema de coordenadas, así que iniciando desde este punto, tenemos que:
Un sistema de coordenadas es un sistema que utiliza uno o más números (coordenadas) para determinar unívocamente la posición de un
punto u objeto geométrico.
El orden en que se escriben las coordenadas es significativo y a veces se las identifica por su posición en una tupla ordenada; también se las
puede representar con letras, como por ejemplo (la coordenada-x). El estudio de los sistemas de coordenadas es objeto de la geometría
analítica, permite formular los problemas geométricos de forma "numérica“.
Teniendo esto en claro, podemos definir a aquello que se conoce como Transformación de coordenadas… Entonces, tenemos que:
La transformación de coordenadas es una operación por la cual una relación, expresión o figura se cambia en otra siguiendo una ley dada.
Analíticamente, la ley se expresa por una o mas ecuaciones llamadas ecuaciones de transformación.
También se define como el cambio de posición de los ejes de referencia en un sistema de coordenadas, ya sea por traslación, rotación, o ambas. El propósito de dicho cambio por lo general es simplificar la ecuación de una curva para manejo posterior.
TRANSFORMACIÓN DE COORDENADAS RECTANGULARES A POLARES
Primero definiremos a cada sistema de coordenadas…
Coordenadas Rectangulares:son aquellas que nos permiten determinar la ubicación de un punto mediante dos distancias y refiriéndolas a una dirección base y a un punto base.
The document discusses coordinate transformations between rectangular and polar coordinate systems. It provides the key formulas for converting between the two systems using trigonometric relationships. Specifically, it gives the formulas to transform from rectangular to polar coordinates as x=rcosθ and y=rsinθ, and from polar to rectangular as x=rcosθ and y=rsinθ, where r is the distance from the origin and θ is the angle relative to the x-axis. It also discusses the process of translating and rotating coordinate axes.
Erika vidal 20% transformacion de coordenadaserikavidal14
This document discusses transformations of coordinate systems, including translation and rotation of axes. It provides examples of converting between rectangular and polar coordinates. Specifically:
1) It explains how to convert between rectangular (x,y) and polar (r,θ) coordinates using trigonometry.
2) Examples are given of converting coordinate points between rectangular and polar formats.
3) Translation of axes is defined as changing the reference axes without rotation, keeping each axis parallel to its original position.
4) Rotation of axes changes the orientation of the coordinate plane while preserving the geometric shape of curves.
The document discusses coordinate transformations, including:
- Converting between rectangular (x,y) and polar (r,θ) coordinate systems using trigonometry and the Pythagorean theorem.
- Translating coordinate axes by shifting the origin point without rotating the axes.
- Rotating coordinate axes by fixing the origin and rotating the x and y axes by an angle using trigonometric formulas.
- Examples are provided of transforming points between rectangular and polar coordinates and translating and rotating coordinate axes. The purpose is to simplify equations for further analysis.
The document introduces concepts of vector algebra and geometry in two dimensions. It defines vectors, vector operations like addition and scalar multiplication, and properties including equality of vectors, parallelism, orthogonality, and the Cauchy-Schwarz inequality. It also covers vector representations geometrically as arrows, vector length, unit vectors, and linear combinations of vectors. Key concepts are illustrated with diagrams of vectors in the Cartesian plane.
This document defines key concepts related to lines in the Euclidean plane including:
1. The definition of a line L as the set of points P0 + ta, where P0 is a base point, a is a non-zero direction vector, and t is a real parameter.
2. Methods for finding the equation of a line including the vector form, parametric form, symmetric form, normal form, and point-slope form.
3. Concepts such as the angle of inclination and slope of a line, and conditions for parallelism and orthogonality between lines.
This document contains a series of exercises related to vectors in a plane. It begins with exercises involving vector operations like finding scalar multiples that satisfy equations and vector addition and subtraction. Later questions involve vector properties such as parallelism of vectors, orthogonality, vector lengths, and linear combinations of vectors. Geometric representations of vectors are also explored through problems finding points and line segments. The document aims to reinforce concepts of vector algebra and geometry through multiple practice problems.
The document defines key concepts in analytic geometry including:
- The Cartesian plane as two perpendicular number lines intersecting at the origin.
- Equations for circles, ellipses, parabolas and hyperbolas in Cartesian coordinates.
- How to find the midpoint between two points in the plane by taking the average of the x- and y-coordinates.
- Formulas for calculating the distance between two points using the Pythagorean theorem.
Slope describes the steepness of a line and is defined as the rise over the run between two points on a line. It can be positive, negative, zero, or undefined. To determine the slope of a line, two points are identified and the change in y-values (rise) is divided by the change in x-values (run). The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. To graph a line using slope-intercept form, the y-intercept is plotted and then additional points are determined by applying the slope using rise-over-run from the previous point.
The document discusses key concepts related to the Cartesian plane including:
- The Cartesian plane is formed by two perpendicular number lines called the x-axis and y-axis intersecting at the origin point.
- Any point P on the plane can be located using its coordinates (x,y) which indicate the point's position along the x and y axes.
- The distance between two points P1(x1,y1) and P2(x2,y2) can be calculated using the distance formula.
- Key curves that can be represented on the Cartesian plane include lines, circles, parabolas, ellipses, and hyperbolas through their defining equations.
This document discusses parametric equations and provides examples of parametric equations for common curves. Specifically, it:
1) Defines parametric equations as equations where the variables x and y are each expressed separately as functions of the same third variable t.
2) Gives parametric equations for circles as x = a cosθ and y = a sinθ, where a is the radius and θ is the angle.
3) Derives the parametric equations for a cycloid as x = r(θ - sinθ) and y = r(1 - cosθ), where r is the radius of the rolling circle and θ is the angle rotated.
3) Discusses eliminating the parameter to
This document discusses different types of equations for geometric shapes and concepts in the coordinate plane. It provides equations for lines, including vector and parametric forms. It also gives the equations for circles, parabolas, ellipses, hyperbolas, and conic sections. Examples are included for finding the midpoint between two points and the distance between two points. The document serves as a reference for the key equations involved in analytic geometry.
The document describes key concepts related to the Cartesian plane including:
- The Cartesian plane consists of two perpendicular axes (x and y) intersecting at the origin point.
- Points on the plane are represented as ordered pairs (x,y).
- The distance between two points P1(x1,y1) and P2(x2,y2) is given by the formula d = √(x2 - x1)2 + (y2 - y1)2.
- Circles, parabolas, ellipses, and hyperbolas are examples of curves that can be represented on the Cartesian plane using algebraic equations. Their properties and equations are discussed.
The branch of mathematics which deals with location of objects in 2-D (dimensional) plane is called coordinate geometry. Need to present your work in most impressive & informative manner i.e. through Power Point Presentation call us at skype Id: kumar_sukh79 or mail us: clintech2011@gmail.com for using my service.
Paso 3 funciones, trigonometría e hipernometría valeria bohorquezjhailtonperez
The document discusses several topics in mathematics including:
1. Cartesian coordinates which use two perpendicular axes (x and y) to locate points on a plane.
2. Venn diagrams which show relationships between sets using circles.
3. Functions which map each element in the domain to a single element in the range.
4. Trigonometry which studies relationships between sides and angles of triangles using trigonometric functions like sine, cosine, and tangent.
The document discusses various concepts in coordinate geometry including:
(1) Calculating the gradient of a straight line from its equation in slope-intercept form or given two points on the line. Parallel and perpendicular lines have related gradients.
(2) Finding the midpoint and length of a line segment given two points.
(3) Determining the equation of a straight line given its gradient and a point, or given two points, or from its graph.
(4) Interpreting the x- and y-intercepts from the equation of a straight line.
The document discusses representing points in 3D space using cylindrical and spherical coordinate systems. It explains that cylindrical coordinates extend the 2D polar coordinate system (r,θ) by adding a z-coordinate. Spherical coordinates represent points using three values - the radial distance ρ, the azimuthal angle θ, and the polar angle φ. The document also covers converting between rectangular and cylindrical/spherical coordinates, and discusses how the elements of integration change based on the coordinate system used.
This document provides information on various concepts in elementary and additional mathematics including:
- The distance, midpoint, and gradient formulas for lines
- Equations of lines
- Parallel and perpendicular lines
- Intersecting lines and finding intersection points
- Perpendicular bisectors
- Finding the area of polygons
It includes examples of applying these concepts to solve problems involving lines, midpoints, gradients, intersections, perpendiculars, and calculating areas.
The document discusses parabolas and their key properties:
- A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- The vertex is the point where the axis of symmetry intersects the parabola. The focus and directrix are a fixed distance (p) from the vertex.
- The latus rectum is the line segment from the focus to the parabola, perpendicular to the axis of symmetry. Its length is determined by the equation of the parabola.
This document provides an overview of conic sections including circles, ellipses, hyperbolas, and parabolas. It discusses how conic sections are formed by the intersection of a double right cone and a plane. Examples are provided on graphing conic sections on a calculator and identifying their properties such as center, vertices, and intercepts. The document also covers using the midpoint and distance formulas to find the center and radius of a circle from its diameter endpoints.
Coordinate systems (and transformations) and vector calculus garghanish
The document discusses various coordinate systems and vector calculus concepts. It defines Cartesian, cylindrical, and spherical coordinate systems. It describes how to write vectors and relationships between components in different coordinate systems. It also covers vector operations like gradient, divergence, curl, and Laplacian as well as line, surface, and volume integrals. Examples are provided to illustrate calculating the gradient of scalar fields defined in different coordinate systems.
The document discusses key concepts in Cartesian geometry including:
- The Cartesian plane is defined by two perpendicular number lines called the x-axis and y-axis intersecting at the origin point.
- Given two points P1(x1,y1) and P2(x2,y2) on the Cartesian plane, the midpoint formula can be used to find the point Pm that is equidistant from both points.
- Common curves that can be represented on the Cartesian plane include circles, parabolas, ellipses, and hyperbolas through their standard equation forms.
The document discusses conic sections, which are curves formed by the intersection of a plane and a right circular cone. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Conic sections can be represented by second-degree equations in x and y, and the technique of completing the square is used to determine which equation corresponds to each type of conic section. The document also reviews the distance formula.
The document discusses calculating the area of a region bounded by a polar curve. It explains that the area can be approximated using Riemann sums of sectors of circles, where the radius of each sector is given by the polar curve. The exact area is given by the limit of these Riemann sums as the number of sectors approaches infinity. The formula for the area of a region bounded by a polar curve r=f(θ) from θ=α to θ=β is ∫_α^β 1/2 r^2 dθ. Several examples are worked out applying this formula to find the area of regions defined by polar equations.
The document discusses various types of coordinate transformations including:
- Converting between rectangular and polar coordinates using trigonometric relationships.
- Examples of transforming points between rectangular and polar coordinates.
- Translating and rotating coordinate axes to simplify equations.
- Representing circles and parabolas using polar coordinate equations.
The document discusses the transformation of coordinates from rectangular to polar coordinates and vice versa. It provides definitions and examples of how to perform these transformations using trigonometric functions. It also explains how to perform translations and rotations of coordinate axes, providing examples of transforming equations under these changes of coordinates. Finally, it discusses representing a circle and parabola using polar coordinate equations.
This document defines key concepts related to lines in the Euclidean plane including:
1. The definition of a line L as the set of points P0 + ta, where P0 is a base point, a is a non-zero direction vector, and t is a real parameter.
2. Methods for finding the equation of a line including the vector form, parametric form, symmetric form, normal form, and point-slope form.
3. Concepts such as the angle of inclination and slope of a line, and conditions for parallelism and orthogonality between lines.
This document contains a series of exercises related to vectors in a plane. It begins with exercises involving vector operations like finding scalar multiples that satisfy equations and vector addition and subtraction. Later questions involve vector properties such as parallelism of vectors, orthogonality, vector lengths, and linear combinations of vectors. Geometric representations of vectors are also explored through problems finding points and line segments. The document aims to reinforce concepts of vector algebra and geometry through multiple practice problems.
The document defines key concepts in analytic geometry including:
- The Cartesian plane as two perpendicular number lines intersecting at the origin.
- Equations for circles, ellipses, parabolas and hyperbolas in Cartesian coordinates.
- How to find the midpoint between two points in the plane by taking the average of the x- and y-coordinates.
- Formulas for calculating the distance between two points using the Pythagorean theorem.
Slope describes the steepness of a line and is defined as the rise over the run between two points on a line. It can be positive, negative, zero, or undefined. To determine the slope of a line, two points are identified and the change in y-values (rise) is divided by the change in x-values (run). The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. To graph a line using slope-intercept form, the y-intercept is plotted and then additional points are determined by applying the slope using rise-over-run from the previous point.
The document discusses key concepts related to the Cartesian plane including:
- The Cartesian plane is formed by two perpendicular number lines called the x-axis and y-axis intersecting at the origin point.
- Any point P on the plane can be located using its coordinates (x,y) which indicate the point's position along the x and y axes.
- The distance between two points P1(x1,y1) and P2(x2,y2) can be calculated using the distance formula.
- Key curves that can be represented on the Cartesian plane include lines, circles, parabolas, ellipses, and hyperbolas through their defining equations.
This document discusses parametric equations and provides examples of parametric equations for common curves. Specifically, it:
1) Defines parametric equations as equations where the variables x and y are each expressed separately as functions of the same third variable t.
2) Gives parametric equations for circles as x = a cosθ and y = a sinθ, where a is the radius and θ is the angle.
3) Derives the parametric equations for a cycloid as x = r(θ - sinθ) and y = r(1 - cosθ), where r is the radius of the rolling circle and θ is the angle rotated.
3) Discusses eliminating the parameter to
This document discusses different types of equations for geometric shapes and concepts in the coordinate plane. It provides equations for lines, including vector and parametric forms. It also gives the equations for circles, parabolas, ellipses, hyperbolas, and conic sections. Examples are included for finding the midpoint between two points and the distance between two points. The document serves as a reference for the key equations involved in analytic geometry.
The document describes key concepts related to the Cartesian plane including:
- The Cartesian plane consists of two perpendicular axes (x and y) intersecting at the origin point.
- Points on the plane are represented as ordered pairs (x,y).
- The distance between two points P1(x1,y1) and P2(x2,y2) is given by the formula d = √(x2 - x1)2 + (y2 - y1)2.
- Circles, parabolas, ellipses, and hyperbolas are examples of curves that can be represented on the Cartesian plane using algebraic equations. Their properties and equations are discussed.
The branch of mathematics which deals with location of objects in 2-D (dimensional) plane is called coordinate geometry. Need to present your work in most impressive & informative manner i.e. through Power Point Presentation call us at skype Id: kumar_sukh79 or mail us: clintech2011@gmail.com for using my service.
Paso 3 funciones, trigonometría e hipernometría valeria bohorquezjhailtonperez
The document discusses several topics in mathematics including:
1. Cartesian coordinates which use two perpendicular axes (x and y) to locate points on a plane.
2. Venn diagrams which show relationships between sets using circles.
3. Functions which map each element in the domain to a single element in the range.
4. Trigonometry which studies relationships between sides and angles of triangles using trigonometric functions like sine, cosine, and tangent.
The document discusses various concepts in coordinate geometry including:
(1) Calculating the gradient of a straight line from its equation in slope-intercept form or given two points on the line. Parallel and perpendicular lines have related gradients.
(2) Finding the midpoint and length of a line segment given two points.
(3) Determining the equation of a straight line given its gradient and a point, or given two points, or from its graph.
(4) Interpreting the x- and y-intercepts from the equation of a straight line.
The document discusses representing points in 3D space using cylindrical and spherical coordinate systems. It explains that cylindrical coordinates extend the 2D polar coordinate system (r,θ) by adding a z-coordinate. Spherical coordinates represent points using three values - the radial distance ρ, the azimuthal angle θ, and the polar angle φ. The document also covers converting between rectangular and cylindrical/spherical coordinates, and discusses how the elements of integration change based on the coordinate system used.
This document provides information on various concepts in elementary and additional mathematics including:
- The distance, midpoint, and gradient formulas for lines
- Equations of lines
- Parallel and perpendicular lines
- Intersecting lines and finding intersection points
- Perpendicular bisectors
- Finding the area of polygons
It includes examples of applying these concepts to solve problems involving lines, midpoints, gradients, intersections, perpendiculars, and calculating areas.
The document discusses parabolas and their key properties:
- A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
- The vertex is the point where the axis of symmetry intersects the parabola. The focus and directrix are a fixed distance (p) from the vertex.
- The latus rectum is the line segment from the focus to the parabola, perpendicular to the axis of symmetry. Its length is determined by the equation of the parabola.
This document provides an overview of conic sections including circles, ellipses, hyperbolas, and parabolas. It discusses how conic sections are formed by the intersection of a double right cone and a plane. Examples are provided on graphing conic sections on a calculator and identifying their properties such as center, vertices, and intercepts. The document also covers using the midpoint and distance formulas to find the center and radius of a circle from its diameter endpoints.
Coordinate systems (and transformations) and vector calculus garghanish
The document discusses various coordinate systems and vector calculus concepts. It defines Cartesian, cylindrical, and spherical coordinate systems. It describes how to write vectors and relationships between components in different coordinate systems. It also covers vector operations like gradient, divergence, curl, and Laplacian as well as line, surface, and volume integrals. Examples are provided to illustrate calculating the gradient of scalar fields defined in different coordinate systems.
The document discusses key concepts in Cartesian geometry including:
- The Cartesian plane is defined by two perpendicular number lines called the x-axis and y-axis intersecting at the origin point.
- Given two points P1(x1,y1) and P2(x2,y2) on the Cartesian plane, the midpoint formula can be used to find the point Pm that is equidistant from both points.
- Common curves that can be represented on the Cartesian plane include circles, parabolas, ellipses, and hyperbolas through their standard equation forms.
The document discusses conic sections, which are curves formed by the intersection of a plane and a right circular cone. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Conic sections can be represented by second-degree equations in x and y, and the technique of completing the square is used to determine which equation corresponds to each type of conic section. The document also reviews the distance formula.
The document discusses calculating the area of a region bounded by a polar curve. It explains that the area can be approximated using Riemann sums of sectors of circles, where the radius of each sector is given by the polar curve. The exact area is given by the limit of these Riemann sums as the number of sectors approaches infinity. The formula for the area of a region bounded by a polar curve r=f(θ) from θ=α to θ=β is ∫_α^β 1/2 r^2 dθ. Several examples are worked out applying this formula to find the area of regions defined by polar equations.
The document discusses various types of coordinate transformations including:
- Converting between rectangular and polar coordinates using trigonometric relationships.
- Examples of transforming points between rectangular and polar coordinates.
- Translating and rotating coordinate axes to simplify equations.
- Representing circles and parabolas using polar coordinate equations.
The document discusses the transformation of coordinates from rectangular to polar coordinates and vice versa. It provides definitions and examples of how to perform these transformations using trigonometric functions. It also explains how to perform translations and rotations of coordinate axes, providing examples of transforming equations under these changes of coordinates. Finally, it discusses representing a circle and parabola using polar coordinate equations.
This document discusses scalar and vector quantities. It defines scalars as quantities that have magnitude but no direction, such as mass, time, length, etc. Vectors are quantities that have both magnitude and direction, and include displacement, velocity, acceleration. The document explains how to represent vectors graphically as arrows and describes different methods to calculate the resultant of concurrent vectors, including the triangle method and polygon method. It also introduces decomposing vectors into rectangular components to calculate vector addition analytically.
The coordinate plane is formed by intersecting two number lines, called the x-axis and y-axis, at their zero points. The point of intersection is called the origin. To graph an inequality in two variables, graph the boundary curve and shade the region where the inequality is true. The distance formula can be used to find the distance between two points by treating it as the hypotenuse of a right triangle formed by the differences in the x and y coordinates. The midpoint formula finds the point halfway between two points by averaging the x and y coordinates. A circle is defined as all points equidistant from a center point, where the distance from the center is the radius. The standard form of a circle equation relates the
Cylindrical and Spherical Coordinates SystemJezreel David
This document discusses cylindrical and spherical coordinate systems. It provides objectives for understanding these coordinate systems, converting between them and Cartesian coordinates, and developing problem-solving skills. Examples are given of converting between cylindrical and Cartesian coordinates, as well as spherical and Cartesian coordinates. Key aspects of cylindrical coordinates include representing points as (r,θ,z) and using conversion equations. Spherical coordinates represent points as (ρ,φ,θ) similar to latitude and longitude. Conversion equations are also provided between the cylindrical and spherical systems.
The polar coordinate system represents each point on a plane using a distance (radial coordinate r) from a fixed point (pole) and an angle (angular coordinate φ) from a fixed direction (polar axis). While polar coordinates are not inherently unique, conventions define r as non-negative and φ within a range of 360° or 2π radians to ensure a unique representation for each point. Polar equations define curves as r as a function of φ, and different forms of symmetry can be deduced from the polar equation. Common curves like circles, lines, roses, spirals, and lemniscates have relatively simple polar equations compared to their Cartesian forms.
The document discusses key concepts related to the Cartesian plane including:
- The Cartesian plane uses two perpendicular number lines (horizontal and vertical) that intersect at the origin point to describe the position of any other point in the plane using coordinates.
- The distance formula can be used to calculate the distance between any two points given their coordinates.
- Important equations are discussed for lines, circles, ellipses, hyperbolas, and parabolas including their standard/canonical forms in the Cartesian plane.
- Methods for graphically representing these conic sections based on their equations are also covered.
This document provides an overview of polar coordinates and graphs. Some key points:
1) Polar coordinates represent points as (r, θ) where r is the distance from the origin and θ is the angle from the positive x-axis.
2) Points are plotted by first finding the angle θ then moving a distance of r units along the terminal side.
3) Formulas are provided to convert between polar and Cartesian coordinates.
4) Various types of curves can be represented using polar equations such as cardioids, limacons, lemniscates, and roses.
5) Symmetry properties of polar graphs are discussed.
This document defines and explains key concepts in analytic geometry including:
- The Cartesian plane consisting of perpendicular x and y axes intersecting at the origin.
- Distances between points on the plane and formulas to calculate distances.
- Midpoint of a segment and properties of circles like radius, diameter, and equations of circles.
- Elements and equations of parabolas, ellipses, and hyperbolas including vertices, foci, axes, and canonical forms.
- René Descartes is credited with developing analytic geometry which uses the Cartesian plane.
The document describes different types of curves and surfaces that can be represented on the Cartesian plane, along with methods for plotting them. It discusses plotting points, lines, circles, parabolas, ellipses, hyperbolas, and their geometric definitions and properties. Various techniques are provided for tracing each curve type given certain parameters like foci, vertices, directors, etc. The Cartesian plane is established as a useful system for locating points and calculating distances between them.
The document discusses the Cartesian coordinate plane and its components. It defines the x-axis and y-axis, the origin point, quadrants, and coordinates. It also explains how to find the distance between two points using their coordinates. Finally, it provides information about circles, parabolas, and ellipses, including their definitions, key elements, and equations.
Chapter 12 vectors and the geometry of space mergedEasyStudy3
This document discusses vectors and geometry in 3D space. It covers topics like 3D coordinate systems, vectors, dot and cross products, equations of lines and planes, cylinders and quadric surfaces. There are also tables listing examples of quadric surface graphs. The document provides information on representing and analyzing geometric objects in 3D space using vectors and coordinate systems.
The document discusses trigonometric functions and radians. It defines a radian as the angle subtended by an arc of a circle that is equal to the radius. A full circle is equal to 2π radians. The trigonometric functions can be defined using a unit circle in radians, allowing the description of periodic processes. Conversion between degrees and radians is covered. Polar coordinates are introduced as an alternative to Cartesian coordinates using radial distance and angle. Trigonometric identities and inverse functions are also discussed.
Trigonometric Function of General Angles LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Trigonometric Functions of Angles
Trigonometric Function Values
Could find the Six Trigonometric Functions
Learn the signs of functions in different Quadrants
Could easily determine the signs of each Trigonometric Functions
Solve problems involving Quadrantal Angles
Find Coterminal Angles
Learn to solve using reference angle
Solve problems involving Trigonometric Functions of Common Angles
Solve problems involving Trigonometric Functions of Uncommon Angles
This document provides an overview of algebra, trigonometry, and analytic geometry. It defines key concepts like functions, coordinate systems, Venn diagrams, and trigonometric functions. Functions are introduced as relations where each element of the domain corresponds to one and only one element in the range. Coordinate systems like the Cartesian plane are explained. Trigonometric functions like sine, cosine, and tangent are defined based on right triangles and the unit circle. Their domains and ranges are described along with periodic properties. Examples of trigonometric and other function types are also given.
The document discusses the Cartesian plane and some of its key elements and uses in geometry. It defines the Cartesian plane as two perpendicular number lines that intersect at an origin point. It describes the axes, quadrants, coordinates, and how geometric shapes like circles and parabolas can be analyzed mathematically using the Cartesian plane. Circles are defined by a center point and radius, and their equations in the Cartesian plane are provided. Properties of parabolas and hyperbolas such as their foci, vertices, and equations are also outlined.
The document discusses the Cartesian plane and its key elements. It describes how René Descartes originated the Cartesian plane by constructing two perpendicular number lines intersecting at a point. The document then defines the key parts of the Cartesian plane, including the x and y axes, quadrants, coordinates, and origin point. It also provides equations for circles, ellipses, hyperbolas, and parabolas in the Cartesian plane.
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1. INSTITUTO UNIVERSITARIO POLITÉCNICO
“SANTIAGO MARIÑO “
AMPLIACIÓN MARACAIBO
DIVISIÓN DE ADMISIÓN Y
CONTROL DE ESTUDIOS
Profesor: Ely Ramírez
UNIDAD 2
Realizadopor:Yuxnei Mora
Cedula:30.757.055
Sección:2-D
Escuela:IngenieríaenPetróleo
Maracaibo, noviembre 2021
2. Desarrollo
1.- Definición y concepto básico de la transformación de coordenadas
Sistemas de Coordenadas] Se refiere a la conversión de un sistema de
coordenadas no proyectado a un sistema de coordenadas utilizando una
serie de ecuaciones matemáticas
2.-Explique cómo se transforman las coordenadas rectangulares a
polares
Coordenadas rectangulares a polares
Las coordenadas polares son escritas de la forma (r, θ), en donde, r es la
distancia y θ es el ángulo. Estas coordenadas pueden ser relacionadas con
las coordenadas rectangulares o cartesianas usando trigonometría, un
triángulo rectángulo y el teorema de Pitágoras. Resulta que usamos la
función tangente para encontrar al ángulo y el teorema de Pitágoras para
encontrar a la distancia, r. A continuación, conoceremos las fórmulas que
podemos usar para transformar de coordenadas rectangulares a polares.
También, resolveremos algunos ejercicios de práctica para aplicar las
fórmulas aprendidas.
Cómo transformar de coordenadas rectangulares a
coordenadas polares
Recordamos que las coordenadas rectangulares son escritas de la forma (x,
y) y las coordenadas polares son escritas de la forma (r, theta), en donde, r
es la distancia desde el origen hasta el punto y θ es el ángulo formado por la
línea y el eje x. Estas coordenadas son relacionadas usando trigonometría.
3. Observemos el siguiente diagrama:
Coordenadas polares 1
Usando el triángulo rectángulo, podemos obtener relaciones para las
coordenadas polares en términos de las coordenadas rectangulares.
Observamos que las coordenadas en x forman la base del triángulo
rectángulo y las coordenadas en y forman la altura. Además, vemos que la
distancia r corresponde a la hipotenusa del triángulo. Entonces, podemos
usar el teorema de Pitágoras para encontrar la longitud de la hipotenusa:
4. El ángulo θ puede ser encontrado usando la función tangente. Recordemos
que la tangente de un ángulo es igual al lado opuesto dividido por el lado
adyacente. El lado opuesto es el componente y y el lado adyacente es el
componente x. Entonces, tenemos:
Debido a que el rango de la función tangente inversa va desde -frac{pi}{2}
hasta frac{pi}{2}, esto no cubre los cuatro cuadrantes del plano cartesiano,
por lo que muchas veces, la calculadora puede dar el valor incorrecto de
{{tan}^{-1}}. Esto depende en el cuadrante en el que se ubica el punto.
Podemos usar lo siguiente para arreglar esto:
Cuadrante Valor de tan-1
I Usamos el valor de la
calculadora
II Sumamos 180° al valor de
la calculadora
III Sumamos 180° al valor de
la calculadora
IV Sumamos 360° al valor de
la calculadora
3.- Explique cómo se transforman las coordenadas polares a
rectangulares
Coordenadas polares a rectangulares
Las coordenadas polares son definidas usando la distancia, r, y al ángulo, θ.
Por otra parte las coordenadas rectangulares, también conocidas como
coordenadas cartesianas, son definidas por x y por y. Podemos encontrar
5. ecuaciones que relacionen a estas coordenadas usando un triángulo
rectángulo y las funciones trigonométricas seno y coseno. A continuación,
conoceremos las fórmulas que podemos usar para transformar de
coordenadas polares a rectangulares. Luego, aplicaremos estas fórmulas al
resolver algunos ejercicios de práctica.
Cómo transformar de coordenadas polares a coordenadas
rectangulares
Las coordenadas polares tienen la forma (r, theta), en donde, r es la
distancia del punto desde el origen y θ es el ángulo formado por la línea y el
eje x. Las coordenadas rectangulares o coordenadas cartesianas tienen la
forma (x, y). Para transformar de coordenadas polares a coordenadas
rectangulares, usamos trigonometría y relacionamos a estas dos
coordenadas.
Consideremos el siguiente diagrama:
6. Claramente, vemos que podemos encontrar las coordenadas x usando la
función coseno y podemos encontrar las coordenadas en y usando la función
seno. Entonces, tenemos las fórmulas:
4.- De un ejemplo para cada una de las transformaciones anteriores
Coordenadas rectangulares a polares resueltos
Lo aprendido sobre la transformación de coordenadas rectangulares a
coordenadas polares es usado para resolver los siguientes ejercicios. Intenta
resolver los ejercicios tú mismo antes de mirar la respuesta.
EJERCICIO 1
Si es que tenemos las coordenadas rectangulares (3, 4), ¿cuál es su
equivalente en coordenadas polares?
Solución
Tenemos los valores x=3, ~y=4. Usamos las fórmulas dadas arriba junto con
estos valores para encontrar las coordenadas polares. Entonces, el valor de r
es encontrado usando el teorema de Pitágoras:
Ejercicio
Ahora, encontramos el valor de θ usando la tangente inversa:
Ejercicio
Tanto el componente en x como el componente en y son positivos, por lo que
el punto está en el primer cuadrante. Esto significa que el ángulo obtenido es
el correcto.
Las coordenadas polares son (5, 0.93 rad).
7. Ejercicio Coordenadas polares a rectangulares
Los siguientes ejercicios son resueltos aplicando las fórmulas de
transformación de coordenadas polares a coordenadas rectangulares.
Intenta resolver los ejercicios tú mismo antes de mirar la respuesta.
EJERCICIO 1
Si es que tenemos a un punto con las coordenadas polares 5,𝜋/3
¿Cuáles son sus coordenadas rectangulares?
Solución
Podemos observar los valores r=5 y 𝜃=𝜋/3. Usamos las fórmulas
encontradas anteriormente para convertir a coordenadas rectangulares.
Entonces, el valor de x es encontrado usando la función coseno:
El valor de y es encontradousando la funciónseno:
y=5(0.866)
y=4.33
Entonces,las coordenadas rectangulares son (2.5, 4.33).
8. 5.- Explique cómo se realiza la traslación de ejes
Espacio para que puedas apreciar el cambio que se produce en las
coordenadas y las expresiones analíticas de una figura cuando se hace una
traslación de ejes.
Para explorar debes:
Para ubicar los nuevos ejes:
a) El eje x se mueve desde el punto B. (haces clic en B y sin soltar lo
cambias de posición)
a) El eje y se mueve desde el punto A. (haces clic en A y sin soltar lo
cambias de posición)
Podrás ver los nuevos ejes, y los valores de las coordenadas nuevas.
El cuadro L.G coordenadas O'X'Y' mostrará la expresión algebraica de la
curva en las nuevas coordenadas.
También puedes mover el punto C y apreciar sus coordenadas en ambos
ejes.
Puedes cambiar los coeficientes de la parábola en el cuadro de dialogo L:G,
6.- Explique cómo se realiza la rotación de ejes
Rotación de ejes
En matemáticas, una rotación de ejes en dos dimensiones es una aplicación
de los puntos de un sistema de coordenadas cartesianas xy sobre los puntos
de un segundo sistema de coordenadas cartesianas denominado x'y', en la
que el origen se mantiene fijo y el los ejes x' e y' se obtienen girando los ejes
x e y en sentido contrario a las agujas del reloj a través de un ángulo 𝜃
9. 7.- Representación Gráfica de una Circunferencia y una Parábola en
Coordenadas polares
CIRCUNFERENCIA
Esta nueva función nos presenta una forma conocida por todos y es
precisamente la circunferencia, la cual será formada en el gráfico polar
mediante la siguiente función:
R=.sen𝜃
10. Ahora veamos una nueva gráfica que resulta en una circunferencia, con la
única diferencia que ahora aparece arriba del rayo inicial (o del eje x que
todos conocemos), a diferencia del gráfico anterior, que la circunferencia
aparecía abajo del radio inicial. La función con su gráfico es esta:
R= 6 sen 𝜃
PARÁBOLA
Esta figura es muy conocida en el mundo del Cálculo. Tal como podemos
generar funciones de parábolas en coordenadas cartesianas, lo podemos
hacer también en coordenadas polares. Veamos el ejemplo: