GEOMETRIC CONSTRUCTION

GEOMETRICCONSTRUCTION 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

GEOMETRY REVIEW • Triangles • Quadrilaterals • Polygons • Circles • Arcs

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

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 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 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 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… BISECTING AN ANGLE TRANSFERRING AN ANGLE

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

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

TRIANGLES… DRAWING A TRIANGLE WITH SIDES GIVEN DRAWING A RIGHT TRIANGLE WITH HYPOTENUSE AND ONE SIDE GIVEN

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

DRAWING AN EQUILATERALTRIANGLE Alternative Method

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 Dividers Method Geometric Method

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 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 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 AutoCAD software provides a convenient object snap for finding tangency.

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

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 ARC TANGENT TO TWO LINES AT ACUTE OROBTUSE ANGLES

DRAWING AN ARC TANGENT TO AN ARC AND A STRAIGHT LINE

DRAWING AN ARC TANGENT TO TWO ARCS

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

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

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

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

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

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 Like cycloids, these curves are used to form the outlines of certain gear teeth and are therefore of practical importance in machine design.

GEOMETRIC CONSTRUCTION