GEOMETRIC CONSTRUCTION CHAPTERFOUR OBJECTIVES 1 Identify the geometry

GEOMETRIC CONSTRUCTION CHAPTERFOUR

OBJECTIVES 1. Identify the geometry that makes up basic 2 D drawings. 2. Use board drafting or 2 D CAD skills to create technical figures. 3. Describe the advantages of CAD contrasted with drawing with manual instruments Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 2 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

GEOMETRY REVIEW • Triangles • Quadrilaterals • Polygons • Circles • Arcs Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 3 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

BISECTING A LINE OR CIRCULAR ARC Compass system Triangle and T-Square System Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 4 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

BISECTING A LINE WITH TRIANGLE AND 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… Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 5 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

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. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 6 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

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. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 7 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

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 Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 8 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

ANGLES… BISECTING AN ANGLE TRANSFERRING AN ANGLE Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 9 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING A LINE PARALLEL TO A LINE AND AT A GIVEN DISTANCE T-square Method Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 10 For Curves © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING A LINE THROUGH A POINT AND PERPENDICULAR TO A LINE When the Point Is Not on the Line Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, When the Point Is on the Line 11 T-square Method © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

TRIANGLES… DRAWING A TRIANGLE WITH SIDES GIVEN DRAWING A RIGHT TRIANGLE WITH HYPOTENUSE AND ONE SIDE GIVEN Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 12 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

LAYING OUT AN ANGLE • Tangent Method • Sine Method • Chord Method Many angles can be laid out directly with the triangle or protractor. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 13 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN EQUILATERAL TRIANGLE Alternative Method Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 14 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING A SQUARE • T-square Method • Diameters Method • Inscribed Circle Method You can use the Auto. CAD Polygon command to draw squares. The Rectangle command is another quick way to make a square in Auto. CAD. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 15 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING A REGULAR PENTAGON Dividers Method Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, Geometric Method 16 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING A HEXAGON Each side of a hexagon is equal to the radius of the circumscribed circle Use a compass Centerline Variation Steps Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 17 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

Drawing an Octagon 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. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 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 18 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

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. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 19 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING TANGENTS TO TWO CIRCLES Auto. CAD software provides a convenient object snap for finding tangency. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 20 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN ARC TANGENT TO A LINE OR ARC AND THROUGH A POINT Tangents Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 21 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN ARC TANGENT TO TWO LINES AT RIGHT ANGLES For small radii, such as 1/8 R 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 Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 22 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN ARC TANGENT TO TWO LINES AT ACUTE OR OBTUSE ANGLES Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 23 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN ARC TANGENT TO AN ARC AND A STRAIGHT LINE Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 24 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN ARC TANGENT TO TWO ARCS Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 25 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

Drawing an Arc Tangent to Two Arcs and Enclosing One or Both Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 26 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN OGEE CURVE Connecting Two Parallel Lines Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, Connecting Two Nonparallel Lines 27 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

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. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 28 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING A FOCI ELLIPSE Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 29 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

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 Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 30 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

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. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 31 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

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. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 32 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

Irregular Curves The curves are largely successive segments of geometric curves, such as the ellipse, parabola, hyperbola, and involute. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 33 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN APPROXIMATE ELLIPSE For many purposes, particularly where a small ellipse is required, use the approximate circular arc method. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 34 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING A PARABOLA The curve of intersection between a right circular cone and a plane parallel to one of its elements is a parabola. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 35 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING A HELIX A helix is generated by a point moving around 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 Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 36 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN INVOLUTE An involute is the path of a point on a string as the string unwinds from a line, polygon, or circle. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 37 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING A CYCLOID A cycloid is generated by a point P on the circumference of a circle that rolls along a straight line Cycloid Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 38 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.

DRAWING AN EPICYCLOID OR A HYPOCYCLOID Like cycloids, these curves are used to form the outlines of certain gear teeth and are therefore of practical importance in machine design. Technical Drawing with Engineering Graphics, 14/e Giesecke, Hill, Spencer, Dygdon, Novak, Lockhart, 39 © 2012, 2009, 2003, Pearson Higher Education, Upper Saddle River, NJ 07458. • All Rights Reserved.