Geometric Construction

Introduction

Strict interpretation of geometric construction allows use of only the compass and an instrument for drawing straight lines, and with these, the geometer, following mathematical theory, accomplishes his solutions. In technical drawing, the principles of geometry are employed constantly, but instruments are not limited to the basic two as T-squares, triangles, scales, curves etc. are used to make constructions with speed and accuracy. Since there is continual application of geometric principles, the methods given in this topic should be mastered thoroughly. It is assumed that students using this book understand the elements of plane geometry and will be able to apply their knowledge.

The constructions given here afford excellent practice in the use of instruments. Remember that the results you obtain will be only as accurate as your skill makes them. Take care in measuring and drawing so that your drawings will be accurate and professional in appearance.

• Geometric Nomenclature

A. Points In Space

A point is an exact location in space or on a drawing surface.

A point is actually represented on the drawing by a crisscross at its exact location. The exact point in space is where the two lines of the crisscross intersect. When a point is located on an existing line, a light, short dashed line or cross bar is placed on the line at the location of the exact point. Never represent a point on a drawing by a dot; except for sketching locations.

B. Line

Lines are straight elements that have no width, but are infinite in length (magnitude), and they can be located by two points which are not on the same spot but fall along the line. Lines may be straight lines or curved lines. A straight line is the shortest distance between two points. It can be drawn in any direction. If a line is indefinite, and the ends are not fixed in length, the actual length is a matter of convenience. If the end points of a line are important, they must be marked by means of small, mechanically drawn crossbars, as described by a pint in space.

Straight lines and curved lines are considered parallel if the shortest distance between them remains constant. The symbol used for parallel line is //. Lines, which are tangent and at 90⁰ are considered perpendicular. The symbol for perpendicular line is ⊥.

C. Angle

An angle is formed by the intersection of two lines. There are three major kinds of angles: right angels, acute angles and

obtuse angles. The right angle is an angle of 90⁰, an acute

Angle is an angle less than 900, and an obtuse angle is an

Angle more than 90⁰, A straight line is 180⁰. The symbol for an angle is < (singular) and <’s (Plural). To draw an angle, use the drafting machine, a triangle, or a protractor.

D. Triangles

A triangle is a closed plane figure with three straight sides and their interior angles sum up exactly 1800. The various kinds of triangles: a right triangle, an equilateral triangle, an isosceles triangle, and an obtuse angled triangle.

E. Quadrialteral

It is a plane figure bounded by four straight sides. When opposite sides are parallel, the quadrilateral is also considered to be a parallelogram.

F. Polygon

A polygon is a closed plane figure with three or more straight sides. The most important of these polygons as they relate to drafting are probably the triangle with three sides, square with four sides, the hexagon with six sides, and the octagon with eight sides.

G. Circle

A circle is a closed curve with all points on the circle at the same distance from the center point. The major components of a circle are the diameter, the radius and circumference.

• The diameter of the circle is the straight distance from one outside curved surface through the center point to the opposite outside curved surface.
• The radius of a circle is the distance from the center point to the outside curved surface. The radius is half the diameter, and is used to set the compass when drawing a diameter.
• A central angle: is an angle formed by two radial lines from the center of the circle.
• A sector: is the area of a circle lying between two radial lines and the circumference.
• A quadrant: is a sector with a central angle of 900 and usually with one of the radial lines oriented horizontally.
• A chord: is any straight line whose opposite ends terminate on the circumference of the circle.
• A segment: is the smaller portion of a circle separated by a chord.
• Concentric circles are two or more circles with a common center point.
• Eccentric circles are two or more circles without a common center point.
• A semi circle is half of the circle.

H. Solids

They are geometric figures bounded by plane surfaces. The surfaces are called faces, and if these are equal regular polygons, the solids are regular polyhedra.

• Techniques Of Geometric Constructions

To construct the above mentioned geometric figures, we have to know some principles and procedures of geometric construction. Thus, the remaining of this chapter is devoted to illustrate step-by-step geometric construction procedures used by drafters and technicians to develop various geometric forms.

A. How To Bisect A Line Or An Arc

To bisect a line means to divide it in half or to find its center point. In the given process, a line will also be constructed at the exact center point at exactly 90⁰.

Given: Line A-B.

Step 1: Set the compass approximately two-thirds of the length of line A-B and swing an arc from point A.

Step 2: Using the exact same compass setting, swing an arc from point B.

Step 3: At the two intersections of these arcs, locate points D and E.

Step 4: Draw a straight-line connecting point D with point E.

Where this line intersects line A-B, it bisects line A-B.

Line D-E is also perpendicular to line A-B at the exact center point.

B. How To Divide A Line In To Number Of Equal Parts

Given: Line A-B.

Step 1: Draw a construction line AC that starts at end A of given line AB. This new line is longer than the given line and makes an angle of not more than 300 with it.

Step 2: Find a scale that will approximately divide the line AB in to the number of parts needed (11 in the example below), and mark these divisions on the line AC.

There are now ‘n’ equal divisions from A to D that lie on the line AC (11 in this example).

Step 3: Set the adjustable triangle to draw a construction line from point D to point B. Then draw construction lines through each of the remaining ‘n-1’ divisions parallel to the first line BD by sliding the triangle along the straight edge. The original line AB will now be accurately divided.

C. How To Bisect An Angle

To bisect an angle means to divide it in half or to cut it in to two equal angles.

Given: Angle BAC.

Step 1: Set the compass at any convenient radius and swing an arc from point A.

Step 2: Locate points E and F on the legs of the angle, and swing two arcs of the same identical length from points E and F, respectively.

Step 3: Where these arcs intersect, locate point D. Draw a straight line from A to D. This line will bisect angle BAC and establish two equal angles: CAD and BAD.

D. How To Draw An Arc Or Circle (Radius) Through Three Given Points

Given: Three points in space at random: A, Band C.

Step 1: With straight line, lightly connect points A to B, and B to C.

Step 2: Using the method outlined for bisecting a line, bisect lines A-B and B-C.

Step 3: Locate point X where the two extended bisectors meet. Point X is the exact center of the arc or circle.

Step 4: Place the point of the compass on point X and adjust the lead to any of the points A, B, or C (they are the same distance), and swing the circle. If all work is done correctly, the arc or circle should pass through each point.

E. How To Draw A Line Parallel To A Straight Line At A Given Distance

Given: Line A-B, and a required distance to the parallel line.

Step 1: Set the compass at the required distance to the parallel line. Place the point of the compass at any location on the given line, and swing a light arc whose radius is the required distance.

Step 2: Adjust the straight edge of either a drafting machine or an adjusted triangle so that it line sup with line A-B, slide the straight edge up or down to the extreme high point, which is the tangent point, of the arc, then draw the parallel line.

F. How To Draw A Line Parallel To A Line Curved Line At A Given Distance

Given: Curved line A-B, and a required distance to the parallel line,

Step 1: Set the compass at the required distance to the parallel line. Starting from either end of the curved line, place the point of the compass on the given line, and swing a series of light arcs along the given line.

Step 2: using an irregular curve, draw a line along the extreme high points of the arcs.

G. How To Draw A Perpendicular Lines To A Line At A Point

Method 1

Given: Line A-B with point P on the same line.

Step 1: Using P as a center, make two arcs of equal radius or more continuous arc (R1) to intercept line A-B on either side of point P, at points S and T.

Step 2: Swing larger but equal arcs (R2) from each of points S and T to cross each other at point U.

Step 3: A line from P to U is perpendicular to line A-B at point P.

H. How To Draw A Perpendicular To A Line At A Point

Method 2

Given: Line A-B with point P on the line.

Step 1: Swing an arc of any convenient radius whose center O is at any convenient location NOT on line A-B, but positioned to make the arc cross line A-B at points P and Q.

Step 2: A line from point Q through center O intercepts the opposite side of the arc at point R.

Step 3: Line R-P is perpendicular to line A-B (A right angle has been inscribed in a semi circle).

I. How To Draw A Perpendicular To A Line From A Point Not On The Line

Given: Line A-B and point P.

Step 1: Using P as a center, swing an arc (R1) to intercept line A-B at points G and H.

Step 2: Swing larger, but equal length arcs (R2) from each of the points G and H to intercept each other at point J.

Step 3: Line P-J is perpendicular to line A-B.

J. How To Draw A Triangle With Known Lengths Of Sides

Given: lengths 1, 2, and 3.

Step 1: Draw the longest length line, in this example length 3, with ends A and B. Swing an arc (R1) from point A whose radius is either length 1 or length 2; in this example length 1.

Step 2; using the radius length not used in step 1, swing an arc (R2) from point B to intercept the arc swung from point A at point.

Step 3: Connect A to C and B to C to complete the triangle.

K. How To Draw A Square

Method-1

Given: The locations of the center and the required distance across the sides of a square.

Step 1: Lightly draw a circle with a diameter equal to the distance around the sides of the square. Set the compass at half the required diameter.

Step 2: Using triangles, lightly complete the square by constructing tangent lines to the circle. Allow the light construction lines to project from the square, with out erasing them.

Step 3: Check to see that there are four equal sides and, if so, darken in the actual square using the correct line thickness.

Method-2

Given one side AB. Through point A, draw a perpendicular.

With A as a center, and AB as radius; draw the arc to intersect the perpendicular at C. With B and C as centers, and AB as radius, strike arcs to intersect at D. Draw line CD and BD.

L. How To Draw A Pentagon (5 Sides)

Given: The locations of the pentagon center and the diameter that will circumscribe the pentagon.

Step 1: Bisect radius OD at C.

Step 2: With C as center, and CA as radius, strike arc AE.

With A as center, and AE as radius, strike arc EB.

Step 3: Draw line AB, then set off distances AB around the circumference of the circle, and draw the sides through these points.

M. How To Draw A Hexagon (6 Sides)

N. To Draw Any Sided Regular Polygon

To construct a regular polygon with a specific number of sides, divide the given diameter using the parallel line method as shown in fig below. In this example, let us assume seven sided regular polygon. Construct an equilateral triangle (0-7-8) with the diameter (0-7) as one of its sides. Draw a line from the apex (point 8) through the second point on the line (point 2). Extend line 8-2 until it intersects the circle at point 9.

Radius 0-9 will be the size of each side of the figure. Using radius 0-9 steps off the corners of the seven sides polygon and connect the points.

O. To Draw A Circle Tangent To A Line At A Given Point

Given: Given line AB and a point on the line.

Step 1: At P erect a perpendicular to the line.

Step 2: Set off the radius of the required circle on the perpendicular.

Step 3: Draw circle with radius CP.

P. To Draw A Tangent To A Circle Through A Point

Method-1

Given: Point P on the circle.

Move the T-square and triangle as a unit until one side of the triangle passes through the point P and the center of the circle; then slide the triangle until the other side passes through point P, and draw the required tangent.

Method-2

Given: Point P outside the circle.

Move the T-square and triangles as a unit until one side of the triangle passes through point P and, by inspection, is the tangent to the circle; and then slide the triangle until the other side passes through the center of the circle, and lightly mark the point of tangency T. finally move the triangle back to its starting position and draw the required tangent.

Q. To Draw Tangents To Two Circles

Move the T-square and triangles as a unit until one side of the triangle is tangent, by inspection, to the two circles; then slide the triangle until the other side passes through the center of one circle, and lightly mark the point of tangency. Then slide the triangle until the side passes through the center of the other circle, and mark the point of tangency. Finally slide the triangle back to the tangent position, and draw the tangent lines between the two points of tangency. Draw the second tangent line in similar manner.

R. How To Construct An Arc Tangent To An Angle

Given: A right angle, lines A and B and a required radius.

Step 1: Set the compass at the required radius and, out of the way, swing a radius from line A and one from line B.

Step 2: From the extreme high points of each radius, construct a light line parallel to line A and another line parallel to line B.

Step 3: Where these lines intersect is the exact location of the required swing point. Set the compass point on the swing point and lightly construct the required radius.

Allow the radius swing to extend past the required area. It is important to locate all tangent points (T.P) before darkening in.

Step 4: Check all work and darken in the radius using the correct line thickness. Darken in connecting straight lines as required. Always construct compass work first, followed by straight lines. Leave all light construction lines.

S. How To Construct An Arc Tangent To Two Radii Or Diameters

Given: Diameter A and arc B with center points located, and the required radius.

Step 1: Set the compass at the required radius and, out of the way, swing a radius of the required length from a point on the circumference of given diameter A. Out of the way, swing a required radius from a point on the circumference of a given arc B.

Step 2: From the extreme high points of each radius, construct a light radius outside of the given radii A and B.

Step 3: Where these arcs intersect is the exact location of the required swing point. Set the compass point on the swing point and lightly construct the required radius.

Allow the radius swing to extend past the required area.

Step 4: Check all work; darken in the radii using the correct line thickness. Darken in the arcs or radii in consecutive order from left to right or from right to left, thus constructing a smooth connecting line having no apparent change in direction.

T. To Draw An Ellipse (By Four-Centered Method)

Join 1 and 3, layoff 3-5 equal to 01-03. This is done graphically as indicated in the fig. Below by swinging 1 around to 5 with O as center where now 03 from 05 is 3-5; the required distance. With 3 as center, an arc from 5 to the diagonal 1-3 locates 6. Bisect 1-6 by a perpendicular crossing

0-1  at 9 and intersecting 0-4 produced (if necessary) at 10.

Make 0-9’ equal to 0-9, and 0-10’ equal to 0-10. Then 9, 9’, 10, and 10’ will be centers for four tangent circle arcs forming a curve approximating the shape of an ellipse.

U. How To Draw An Ogee Curve

An ogee curve is used to join two parallel lines. It forms a gentle curve that reverses itself in a neat symmetrical geometric form.

Given: Parallel lines A-B and C-D.

Step 1: Draw a straight line connecting the space between the parallel lines. In this example, from point B to point C.

Step 2: Make a perpendicular bisector to line B-C to establish point X.

Step 3: Draw a perpendicular from line A-B at point B to intersect the perpendicular bisector of B-X, which locates the first required swing center. Draw a perpendicular from line C-D at point C to intersect the perpendicular bisector of CX, which locates the second required swing center.

Step 4: Place the compass point and adjust the compass lead to point B, and swing an arc from B to X. Place the compass point on the second swing point and swing an arc from X to C. This completes the ogee curve.