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GEOMETRIC CONSTRUCTION. C H A P T E R F O U R. 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.

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geometric construction

GEOMETRICCONSTRUCTION

C H A P T E R F O U R

objectives
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

geometry review
GEOMETRY REVIEW
  • Triangles
  • Quadrilaterals
  • Polygons
  • Circles
  • Arcs
bisecting a line or circular arc
BISECTING A LINE OR CIRCULAR ARC

Compass system

Triangle and T-Square System

bisecting a line with triangle and t square
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…

triangles
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.

any angle in 15 increments
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.

protractors
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

angles
ANGLES…

BISECTING AN ANGLE

TRANSFERRING AN ANGLE

drawing a line through a point and perpendicular to a line
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

triangles12
TRIANGLES…

DRAWING A TRIANGLE WITH SIDES GIVEN

DRAWING A RIGHT TRIANGLE WITH

HYPOTENUSE AND ONE SIDE GIVEN

laying out an angle
LAYING OUT AN ANGLE
  • Tangent Method
  • Sine Method
  • Chord Method

Many angles can be laid out directly with the triangle or protractor.

drawing a square
DRAWING A SQUARE
  • T-square Method
  • Diameters Method
  • Inscribed Circle Method

You can use the AutoCAD Polygon

command to draw squares. The Rectangle command is another quick way to make a square in AutoCAD.

drawing a regular pentagon
DRAWING A REGULAR PENTAGON

Dividers Method

Geometric Method

drawing a hexagon
DRAWING A HEXAGON

Each side of a hexagon is equal to the radius of the circumscribed circle

Use a compass

Centerline Variation

Steps

drawing an octagon
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.

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

finding the center of a circle
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.

drawing tangents to two circles
DRAWING TANGENTS TO TWO CIRCLES

AutoCAD software provides a convenient object snap for finding tangency.

drawing an arc tangent to two lines at right angles
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

drawing an ogee curve
DRAWING AN OGEE CURVE

Connecting Two Parallel Lines

Connecting Two Nonparallel Lines

the conic sections
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.

drawing a concentric circle ellipse
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

drawing a parallelogram ellipse
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.

ellipse templates
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.

irregular curves
Irregular Curves

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

drawing an approximate ellipse
DRAWING AN APPROXIMATE ELLIPSE

For many purposes, particularly where a small ellipse is required, use the approximate circular arc method.

drawing a parabola
DRAWING A PARABOLA

The curve of intersection between a right circular cone and a plane parallel to one of its elements is a parabola.

drawing a helix
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

drawing an involute
DRAWING AN INVOLUTE

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

drawing a cycloid
DRAWING A CYCLOID

A cycloid is generated by a point P on the circumference of a circle that rolls along a straight line

Cycloid

drawing an epicycloid or a hypocycloid
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.