Geometric Constructions for Drawing and Layout: Methods and Techniques

1. CH 7 GEOMETRIC CONSTRUCTIONS

2. LAYING OUT AN ANGLE • Sine Method • Chord Method • Tangent Method Many angles can be laid out directly with the triangle or protractor.

3. DRAWING A SQUARE • T-square Method • Diameters Method • Inscribed Circle Method

4. DRAWING A HEXAGON Each side of a hexagon is equal to the radius of the circumscribed circle Use a compass Centerline Variation Steps

5. DRAWING AN OCTAGON Given a circumscribed square, (the distance “across flats”) draw the diagonals of the square. Then, use the corners of the square as centers and half the diagonal as the radius to draw arcs cutting the sides Given an inscribed circle, or distance “across flats”, use a T-square or straightedge and a 45° triangle to draw the eight sides tangent to the circle.

7. DRAWING AN ARC TANGENT TO A LINE OR ARC AND THROUGH A POINT Tangents

8. DRAWING AN ARC TANGENT TO TWO LINES AT RIGHT ANGLES For small radii, such as 1/8R for fillets and rounds, it is not practicable to draw complete tangency constructions. Instead, draw a 45° bisector of the angle and locate the center of the arc by trial along this line

10. DRAWING AN ARC TANGENT TO TWO LINES AT ACUTE OROBTUSE ANGLES

11. DRAWING AN ARC TANGENT TO AN ARC AND A STRAIGHT LINE

12. DRAWING AN ARC TANGENT TO TWO ARCS

13. DRAWING AN OGEE CURVE Connecting Two Parallel Lines Connecting Two Nonparallel Lines

14. DRAWING A FOCI ELLIPSE

15. DRAWING A CONCENTRIC CIRCLE ELLIPSE If a circle is viewed with the line of sight perpendicular to the plane of the circle… …the circle will appear as a circle, in true size and shape

16. Irregular Curves The curves are largely successive segments of geometric curves, such as the ellipse, parabola, hyperbola, and involute.

17. DRAWING AN APPROXIMATE ELLIPSE For many purposes, particularly where a small ellipse is required, use the approximate circular arc method.

18. DRAWING A HELIX A helix is generated by a point moving around and along the surface of a cylinder or cone with a uniform angular velocity about the axis, and with a uniform linear velocity about the axis, and with a uniform velocity in the direction of the axis

19. DRAWING AN INVOLUTE An involute is the path of a point on a string as the string unwinds from a line, polygon, or circle.

21. GEOMETRIC CONSTRUCTION EXERCISE 5

34. BISECTING A LINE WITH TRIANGLEAND T-SQUARE From endpoints A and B, draw construction lines at 30°, 45°, or 60° with the given line. Then, through their intersection, C, draw a line perpendicular to the given line to locate the center D…

35. BISECTING A LINE OR CIRCULAR ARC Compass system Triangle and T-Square System

36. Triangles Inclined lines can be drawn at standard angles with the 45° triangle and the 30° x 60° triangle. The triangles are transparent so that you can see the lines of the drawing through them. A useful combination of triangles is the 30° x 60° triangle with a long side of 10" and a 45° triangle with each side 8" long.

37. Any Angle in 15° Increments With only a 30° x 60° triangle and a 45° triangle, you can draw any angle in 15° increments The bottom of the triangle in each case is resting on the blade of the T-square. Twenty-four 15° sectors are possible with just these two triangles used singly or in combination.

38. Protractors For measuring or setting off angles other than those obtainable with triangles, use a protractor. Plastic protractors are satisfactory for most angular measurements Nickel silver protractors are available when high accuracy is required

39. DRAWING A LINE PARALLEL TO A LINE AND AT A GIVEN DISTANCE T-square Method For Curves

40. DRAWING A LINE THROUGH A POINT AND PERPENDICULAR TO A LINE When the Point Is Not on the Line When the Point Is on the Line T-square Method

41. ANGLES… BISECTING AN ANGLE TRANSFERRING AN ANGLE

42. DRAWING AN EQUILATERALTRIANGLE Alternative Method

43. DRAWING A REGULAR PENTAGON Dividers Method Geometric Method

44. FINDING THE CENTER OF A CIRCLE This method uses the principle that any right triangle inscribed in a circle cuts off a semicircle. Another method, slightly longer, is to reverse the procedure. Draw two nonparallel chords and draw perpendicular bisectors. The intersection of the bisectors will be the center of the circle.

45. DRAWING TANGENTS TO TWO CIRCLES

46. Drawing an Arc Tangent to Two Arcs and Enclosing One or Both

47. DRAWING A PARALLELOGRAM ELLIPSE The intersection of like-numbered lines will be points on the ellipse. Locate points in the remaining three quadrants in a similar manner. Sketch the ellipse lightly through the points, then darken the final ellipse with the aid of an irregular curve.

48. THE CONIC SECTIONS The conic sections are curves produced by planes intersecting a right circular cone. Four types of curves are produced: the circle, ellipse, parabola, and hyperbola, according to the position of the planes.

49. ELLIPSE TEMPLATES These ellipse guides are usually designated by the ellipse angle, the angle at which a circle is viewed to appear as an ellipse.

50. OBJECTIVES 1. Identify the geometry that makes up basic 2D drawings. 2. Use board drafting or 2D CAD skills to create technical figures. 3. Describe the advantages of CAD contrasted with drawing with manual instruments