Drawing Techniques - Geometric Constructions

Spring 2011, ME102 – Engineering Design GraphicsDr. Jorge Luis Abanto-Bueno

Drawing Techniques

Geometric

Constructions

Lecture 26, Slide 1

Spring 2011, ME102 – Engineering Design GraphicsDr. Jorge Luis Abanto-Bueno Lecture 2, Slide 2

1. Bisecting a LineA. Given line AB of known length L, draw a perpendicular line that passes through

the mid point of the given line AB


1. Bisecting a LineB. Draw two arcs with radius R1 >

0.5xL and centered at points A and B


1. Bisecting a LineC. Draw a line that passes through

the two points were the arcs intersect each other


2. Bisecting an AngleA. Given lines AB and BC that intersect at point B and form angle q, draw a line

that passes through B and bisects angle q

θ


2. Bisecting an AngleB. Draw an arc with radius R1 (> 0) and center at point B. The arc will intersect

line AB at point D and line BC at point E


2. Bisecting an AngleC. Draw two arcs centered at points D

and E and radius equal to R2 > 1.5xR1. These two arcs will intersect at point D


2. Bisecting an AngleD. Draw a line that

starts at point B and passes through point D


3. Tangent to a Circle at a Given PointA. Given a circle and point P on the

circumference of the circle, draw a tangent line to circle that passes through point P


3. Tangent to a Circle at a Given PointB. Draw a line from the center of the

circle that passes through point PC. Draw an arc centered at point B and

any radius (R1 > 0) that will intersect previous line at points A and B


3. Tangent to a Circle at a Given PointD. Draw two arcs centered at

points A and B and radius R2>1.5xR1. The arcs will intersect at point C


3. Tangent to a Circle at a Given PointE. Draw a line that passes

through points C and P


4. Tangent to a Circle from a Given PointA. Given a circle ,centered at point O, and point P, which is not a point of the

circumference of the circle, draw tangent line to circle that passes through point P


4. Tangent to a Circle from a Given PointB. Draw a line that joins points O and P C. Draw two arcs centered at

points O and P and radius R1 > 0.5xOP. These arcs will intersect at points A and B


4. Tangent to a Circle from a Given PointD. Draw a line that joins points

A and B and intersects line OP at point C

E. Draw an arc centered at point C and radius equal to length OC. Arc will intersect the given circle at points D and E


4. Tangent to a Circle from a Given PointF. Draw a line that joints either points

P and D or points P and E.


5. Arc Tangent to two linesA. Given two lines PQ and QR that intersect at point Q,

draw an arc tangent to these two lines with radius R1


B. Select two points on each line such as points C and D (line PQ) and A and B (line QR)

C. Draw arcs centered at these points and radius R1

5. Arc Tangent to two lines


D. Draw two lines tangent to the arcsE. These two lines will intersect at point J

5. Arc Tangent to two lines


5. Arc Tangent to two linesF. Draw an arc centered at point J and radius R1. This

arc will intersect line PQ at point L and line QR at point K


6. S-shaped Curves: 1st MethodA. Given parallel lines QS and PR, draw an S-shaped curve curve that is tangent to line

QS at point C, tangent to line PR at point A, and passes through point B. Points A, B and C form a straight line


6. S-shaped Curves: 1st MethodB. Draw two lines perpendicular to lines QS and PR that pass through points C and A,

respectively (use the bisecting of a line method)


6. S-shaped Curves: 1st MethodC. Draw three arcs centered at points A, B and C using the same radius for all the arcs

These arcs must intersect at points j, l, k and m


6. S-shaped Curves: 1st MethodD. Draw two lines that pass through points l and m, and j and k


6. S-shaped Curves: 1st MethodE. Extend lines lm and jk to intersect the perpendicular lines drew in step B at points

O and N, respectively


6. S-shaped Curves: 1st MethodF. Centered at point N and radius equal to length AN draw an arc that passes through

points A and B


6. S-shaped Curves: 1st MethodG. Centered at point O and radius equal to length OC draw an arc that passes through

points B and C


7. S-shaped Curves: 2st MethodA. Given points A and B on lines QS and PR, respectively,

draw an S-shaped curved with radius R1


7. S-shaped Curves: 2st MethodB. Draw a straight line that passes through points A and BC. Draw lines perpendicular to lines QS and PR at points A

and B, respectively (use the bisecting a line method)


7. S-shaped Curves: 2st MethodD. Draw an arc centered at point A that intersects the line

drawn in step C at point gE. Draw an arc centered at point g that intersects line AB

at point h


7. S-shaped Curves: 2st MethodF. Draw a line that passes through points g and h and intersects

the other perpendicular line (step C) at point iG. Draw an arc centered at point i and radius R2 equal to

distance Bi. Points h and B are the end points of this arc


8. Arc Tangent to a Line and a Circle

A. Given a circle centered at point O and radius R, and line AB, draw an arc tangent to both of them. The radius of the arc tangent must be R1



B. Draw line ef parallel to line AB at a distance equal to radius R1



C. Draw and arc centered at point O and radius R+R1. This arc will intersect line ef at point h

D. Draw a line from point O to point h. This line will intersect the given circle at point g



E. Draw an arc centered at point h and radius R1. This arc will be tangent the given circle and line AB at points g and i, respectively


9. Arc Tangent to a Line and an Arc

A. Given an arc centered at point O and radius R, and line AB, draw an arc tangent to both of them. The radius of the arc tangent must be R1



B. Draw line cd parallel to line AB at a distance equal to radius R1



C. Draw an arc centered at point O and radius equal to R-R1. This arc will intersect line cd at point e.

D. Draw a line that contains points e and O. This line will intersect the given arc at point g



E. Draw an arc centered at point e and radius R1. This arc will be tangent to line AB at point f and tangent to the given arc at point g


10. Concave Arc Tangent to two Arcs(Circles)A. Given two arcs (or circles) centered at points A and

B and radius R1 and R2, respectively, draw a concave arc with radius R and tangent to both of them


10. Concave Arc Tangent to two Arcs(Circles)

B. Draw arcs centered at points A and B and radius R1+R and R2+R. The arcs will intersect at point c


10. Concave Arc Tangent to two Arcs(Circles)C. Draw a line from point A to point c. This line

will intersect the corresponding arc (or circle) at point d

D. Draw a line from point B to point c. This line will intersect the corresponding arc (or circle) at point e


10. Concave Arc Tangent to two Arcs(Circles)

E. Draw an arc centered at point c and radius R. This arc will be tangent to the given arcs (or circles) at points d and e


11. Convex Arc Tangent to two Arcs(Circles)

A. Given two arcs (or circles) centered at points A and B and radius R1 and R2, respectively, draw a convex arc with radius R and tangent to both of them



B. Draw arcs centered at points A and B with radius R-R1 and R-R2, respectively. The arcs will intersect at point a



C. Draw a line that passes through points A and a and intersects the corresponding arc at point b



D. Draw a line that passes through points B and a and intersects the corresponding arc at point c



E. Draw an arc centered at point a and radius R. This arc will be tangent to the given arcs at points c and d


F. Final geometric shape



12. Arc tangent to two circles (or arcs)

A. Given two circles (or arcs) centered at points A and B and radius R1 and R2, respectively, draw an arc with radius R and tangent to both of the given circles (or arcs)


12. Arc tangent to two circles (or arcs)

B. Draw an arc centered at point A and radius R+R1

C. Draw an arc centered at point B and radius R-R2Arcs will intersect at point C


12. Arc tangent to two circles (or arcs)D. Draw a line from point c to point A. This

line will intersect the corresponding circle at point e

E. Draw a line that starts at point c, passes through point B and intersects the corresponding circle at point d


12. Arc tangent to two circles (or arcs)E. Final geometric shape


13. Arc tangent to an Arc and a Circle

A. Given an arc centered at point B and radius R2, and a circle centered at point A and radius R1, draw an arc

tangent to both of them with radius R



B. Draw an arc centered at point A and radius R1+R

C. Draw an arc centered at point B and radius R2-R

Arcs will intersect at point a



D. Draw an arc centered at point a and radius R. This arc will be tangent to the given arc and circle at points c and b, respectively



E. Final geometric shape



A. Given an arc centered at point B and radius R2, and a circle centered at point A and radius R1, draw an arc

tangent to both of them with radius R


B. Draw an arc centered at point A and radius R-R1

C. Draw an arc centered at point B and radius R2-R

Arcs will intersect at point c



D. Draw a line that starts at point B, passes through point c and intersects the arc at point e



E. Draw a line that starts at point c, passes through point A and intersects the circle at point d



F. Draw an arc centered at point c and radius R. The arc will intersect the given arc and circle at points e and d



G. Final geometric shape


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Drawing Techniques - Geometric Constructions