Geometricalconstruction

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Geometricalconstruction

  1. 1. Introduction In the course of engineering drawing, it is often necessary to make a certain geometrical constructions in order to complete an outline. There are no projections involved, and no dimensioning problems, the ONLY GREAT DIFFICULTY IS ACCURACY. Common geometric shapes
  2. 2. PRELIMINARY TECHNIQUES Geometrical Construction Techniques LINES A POINT has no area, it indicates a position, it can be indicated by a dot or thus A LINE has length but no area. It may be curved or straight. A STRAIGHT LINE is the shortest distance between two points. GEOMETRICAL TERMS
  3. 3. BISECTING/PERPENDICULARS/PARALLELS/DIVISION 1) BISECT A LINE 1. With a compass opened to a distance greater than half AB, strike arcs from A and B. 2. A line joining the points of intersection of the arcs is the bisector. 2) BISECT AN ACUTE ANGLE 1. Set an acute angle (angle less than 90:), and bisect the angle. 3) BISECT OF A GIVEN ARC 1. With centre A and radius greater than half AB, describe an arc. 2. Repeat with the same radius from B, the arcs intersecting at C and D. Join C to D to bisect the arc AB. 4) PERPENDICULAR AT A POINT ON A LINE 1. At point O, draw a semicircle of any radius to touch the line at a and b. 2. With compass at a greater radius, strike arcs from a and b. 5) PARALLEL LINE TO A LINE WITH A GIVEN DISTANCE. AB is the given line, C is the given distance. 1. From any two points well apart on AB, draw two arcs of radius equal to C. 2. Draw a line tangential to the two arcs to give required line. 6) DIVISION OF A LINE INTO EQUAL PARTS AB is the given line. 1. Draw a line AC at any angle. 2. On line AC, make three convenient equal divisions. 3. Join the last division with B and draw parallel lines as shown.
  4. 4. CONSTRUCTIONS OF ANGLES TERMINOLOGY . NAMES OF ANGLES 7) CONSTRUCTION OF A 60° AND 30° ANGLES 8) CONSTRUCTION OF A 45° AND 90° ANGLES If two lines are pivoted as shown in the diagram, as one line opens they form an angle. If the rotation is continued the line will cover a full circle. The unit for measuring an angle is a
  5. 5. CONSTRUCTION OF TRIANGLES TERMINOLOGY A triangle is a plane figure bounded by three straight lines. Triangles are named according to the length of their sides or the magnitude of their angles. EQUILATERAL All angles 60°. All sides equal. ISOSCELES Base angles equal. Opposite sides equal RIGHT ANGLE One angle is 90°. All sides of different length. OBTUSE ANGLE One angle is greater than 90°. All sides of different length. SCALENE All angles different. All sides of different length.
  6. 6. CONSTRUCTION OF TRIANGLES 9) TO CONSTRUCT AN EQUILATERAL TRIANGLE 1.Draw a line AB, equal to the length of the side. 2.With compass point on A and radius AB, draw an arc as shown above. 10) TO CONSTRUCT AN ISOSCELES TRIANGLE, GIVEN BASE AND VERTICAL HEIGHT 1.Draw line AB. 2.Bisect AB and mark the vertical height. ABC is the required isosceles triangle. 11) TO CONSTRUCT A RIGHT-ANGLE TRIANGLE 1.Draw AB. From A construct angle CAB. 2.Bisect AB. Produce the bisection to cut AC at O. 3.With centre O and radius OA, draw semi-circle to find C. Complete the triangle 12) TO CONSTRUCT A TRIANGLE, GIVEN THE BASE ANGLES & THE ALTITUDE 1.Draw a line AB. Construct CD parallel to AB so that the distance between them is equal to the latitude. 2.From any point E, on CD, draw CÊF & DÊG so that they cut AB in F & G respectively. 3.Since CÊF = EFG & DÊG = EĜF (alternate angles), then EFG is the required triangle.
  7. 7. THE CIRCLE PARTS OF A CIRCLE 13) TO FIND THE CENTRE OF A GIVEN ARC 1. Draw two chords, AC and BD. 2. Bisect AC and BD as shown. The bisectors will intersect at E. 3. The centre of the arc is point E. 14) TO FIND THE CENTRE OF CIRCLES (METHOD 1) 1. Draw two horizontal lines facing one another across the circle at a place approximately halfway from the top to the centre of the circle. These lines pass through the circle form points A, B, C and D. 2. Bisect these two lines. Where these two bisect lines intersect, thus the centre of the given circle. 15) TO FIND THE CENTRE OF CIRCLES (METHOD 2) 1. Draw a horizontal line across the circle at a place approx. halfway from the top to the centre of the circle. 2. Draw perpendicular lines downward from A and B. Where these lines cross the circle forms C & D. 3. Draw a line from C to B and from A to D. Where these lines cross is the exact centre of the given circle.
  8. 8. QUADRILATERALS TERMINOLOGY The quadrilateral is a plane figure bounded by four straight sides SQUARE All four sides equal. All angles 90:. RECTANGLE Opposite sides of equal. All angle 90:. RHOMBUS All four sides equal. Opposite angles equal. PARALLELOGRAM Opposite sides equal. Opposite angles equal. TRAPEZIUM Two parallel sides. Two pairs of angles equal. 16) TO CONSTRUCT A SQUARE 1.Draw the side AB. From B erect a perpendicular. Mark off the length of side BC. 2.With centres A & C draw arcs, radius equal to the length of the side of the square, to intersect at D. ABCD is the required square. 17) TO CONSTRUCT A PARALLELOGRAM 1.Draw AD equal to the length of one of the sides. From A construct the known angle. Mark off AB equal length to other known side. 2.With compass pt. at B draw an arc equal radius to AD. With compass pt. at D draw an arc equal in radius to AB. ABCD is the required parallelogram. 18) TO CONSTRUCT A RHOMBUS 1.Draw the diagonal AC. 2.From A and C draw intersecting arcs, equal in length to the sides, to meet at B and D. ABCD is the required rhombus.
  9. 9. REGULAR POLYGONS TERMINOLOGY A polygon is a plane figure bounded by more than four straight sides. Regular polygons are named according to the number of their sides. PENTAGON :5 sides HEPTAGON :7 sides NONAGON :9 sides UNDECAGON :11 sides HEXAGON :6 sides OCTAGON :8 sides DECAGON :10 sides DODECAGON :12 sides The regular polygons drawn on this page are the figures most frequently used in geometrical drawing. Particularly the hexagon and the octagon which can be constructed by using 60⁰ or 45⁰ set-square. REGULAR PENTAGON Five sides equal. Five angles equal. REGULAR HEXAGON Six sides equal. Six angles equal. REGULAR OCTAGON Eight sides equal. Eight angles are equal. IRREGULAR PENTAGON Five sides unequal. Five angles unequal. RE-ENTRANT HEXAGON One interior angle greater than 180:. Six sides & six angles unequal. IRREGULAR HEPTAGON Seven sides unequal. Seven angles unequal.
  10. 10. ) TO CONSTRUCT A19 HEXAGON, GIVEN THE DISTANCE ACROSS THE (A/C)CORNERS 1.Draw a vertical and horizontal centre lines and a circle with a diameter equal to the given distance. 2.Step off the radius around the circle to give six equally spaced points, and join the points to give the required hexagon. 20) TO CONSTRUCT A HEXAGON, GIVEN THE DISTANCE ACROSS THE FLATS (A/F) 1.Draw vertical and horizontal centre lines and a circle with a diameter equal to the given distance. Use a 60: set-square and tee-square as shown to give the six sides. 21) TO CONSTRUCT AN OCTAGON, GIVEN THE DISTANCE ACROSS CORNERS (A/C) 1.Draw vertical and horizontal centre lines and a circle with a diameter equal to the given distance. 2.With a 45: set-square, draw points on the circumference 45: apart. Connect these eight points by straight lines to give the required octagon.
  11. 11. ) TO CONSTRUCT AN22 OCTAGON, GIVEN THE (A/C)DISTANCE ACROSS CORNERS 1.Draw vertical and horizontal centre lines and a circle with a diameter equal to the given distance. 2.With a 45: set-square, draw points on the circumference 45: apart. 3.Connect these eight points by straight lines to give the required octagon. ) TO CONSTRUCT AN OCTAGON,23 GIVEN THE DISTANCE (A/FACROSS THE FLATS 1.vertical and horizontal centre lines and a circle with a diameter equal to the given distance. 2.Use a 45: set-square and tee-square as shown in construction of hexagon A/F to give the eight sides. 24) TO INSCRIBE ANY REGULAR POLYGON WITHIN A CIRCLE. e.g. PENTAGON
  12. 12. TANGENTS TO CIRCLES TERMINOLOGY If a disc stands on its edge on a flat surface it will touch the surface at one point. This point is known as the point of tangency as shown in the diagram and the straight line which represents the flat plane is known as a tangent. A line drawn from the point of tangency to the centre of the disc is called normal, and the tangent makes an angle of 90° with the normal.
  13. 13. 25) EXTERNAL TANGENT TO TWO CIRCLES OF DIFFERENT Ø (OPEN BELT) 1. Join the centres of circles a and b. Bisect ab to obtain the centre c of the semicircle. 2. From the outside of the larger circle, subtract the radius r of the smaller circle. Draw the arc of radius ad. Draw normal Na. 3. Normal Nb is drawn parallel to normal Na. Draw the tangent. 26) INTERNAL TANGENT TO TWO CIRCLES OF DIFFERENT Ø (CROSS BELT) 1. Join the centres of circles a and b. Bisect ab to obtain the centre c of the semicircle. 2. From the outside of the larger circle, add the radius r of the smaller circle. Draw the arc of radius ad. Draw normal Na. 3. Normal Nb is drawn parallel to normal Na. Draw the tangent.
  14. 14. JOINING OF CIRCLES 27) OUTSIDE RADIUS Two circles of radii a and b are tangential to arc of radius R. 1. From the centre of circle radius a, describe an arc of R + a. 2. From the centre of circle radius b, describe an arc of R + b. 3. At the intersection of the two arcs, draw arc radius R. 28) INSIDE RADIUS Two circles of radii a and b are tangential to arc of radius R. 1. From the centre of circle radius a, describe an arc of R - a. 2. From the centre of circle radius b, describe an arc of R - b. 3. At the intersection of the two arcs, draw arc radius R.
  15. 15. THE ELLIPSE TERMINOLOGY 29) CONCENTRIC/AUXILIARY CIRCLE METHOD 1.Draw two circles around the major and minor axis. 2.Divide into twelve equal parts using 30: - 60: set-square. 3.Draw horizontal lines from the minor circle and vertical lines from the major circle. 4.The intersection points between horizontal and vertical lines are points of an ellipse.
  16. 16. AN INVOLUTE TERMINOLOGY There are several definitions for the involutes, none being particularly easy to follow. An involute is the path of a point on a string as the string unwinds from a line, polygon, or circle. And it is also the locus of a point, initially on a base circle, which moves so that its straight line distance, along a tangent to the circle, to the tangential point of contact, is equal to the distance along the arc of the circle from the initial point to the instant point of tangency. The involute is best visualized as the path traced out by the end of a piece of cotton when cotton is unrolled from its reel. 30) TO DRAW AN INVOLUTE OF A CIRCLE Let the diameter of the circle is given 1. Divide the circle into 12 equal parts. 2. Draw tangents at each of the twelve circumferential divisions point, setting off along each tangent the length of the corresponding circular arc. 3. Draw the required curve through the points set off and can be determined by setting off equal distances 0-1, 1-2, 2-3, and so on, along the circumference. NOTE: The involutes of a circle are used in the construction of involutes gear teeth. In this system, the involutes form the face and a part of the flank of the teeth of gear wheels; the outlines of the teeth of racks are straight lines.

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