Mathematics

Key unit competence

By the end of this unit, I will be able to transform shapes under orthogonal or parallel projection.

Unit outline

• Definition of parallel projection

• Properties of parallel projection

• Definition of orthogonal projection

• Properties of orthogonal projection

• Images of geometrical shapes under parallel projection and orthogonal projection.

8.1 Parallel projection

8.1.1 Introduction to parallel  projection

Activity 8.1

In this activity, you will draw two parallel line using a ruler and a set square.

Step 1:  Using a ruler, draw a straight horizontal parallel line AB as shown in figure 8.1 below.

Step 2:  Align the base of the set square to the line AB as shown in the figure 8.2.

Step 3: Place the ruler along the set square as shown in figure 8.3.

Step 4:  Slide the set square along the ruler (you can slide it for about 2 cm) as figure 8.4 shows. As you slide up the set square, ensure that you hold the ruler down firmly.

Step 5: Draw a line at the new base of the set square. Is the line parallel to line AB?

Step 6:  Measure the distance between the two lines at different positions. What can you comment about the distance between the two lines?

The distance between the two lines are discovered to be equal. This new line is parallel to the line AB

From activity 8.1, we discovered that parallel lines are those lines which can never intersect. This is because they have a constant distance between them.

8.1.2  Parallel projection of a point on a line

Activity 8.2

1. In your note book draw a line and label it L.

2. On same side of line L, mark two points A and B not more than 3 cm from L, and about 2 cm apart.

3. From point A, draw any-line segment to meet L at point A′. From B, draw another line segment to parallel to AA′ to meet L and B′.

This figure shows a one to one mapping where A′ is the image of A and B′ the image of B on line L

In such a mapping A′ is called the projection of A and B′ the projection of B on line L. This is called a parallel projection. Why is this a parallel projection?

Note

If we join A to B in fig 8.6 would be the parallel projection of line segment AB on the line L.

Shadows are formed as the result of an opaque object placed in front of the source of light ray where parallel light rays project the object to the wall. A shadow is the projection of the object on the wall parallel to the direction of the light rays as shown on the figure 8.7 below.

Consider triangle ABC which is placed in the direction of parallel lines as fig 8.8 shows. The triangle is projected by the parallel lines to form the image along the straight line XY.

A′, B′, and C' show where the vertices are moved by “parallel” projection. Parallel projection of a point can also be described in the following steps.

Let L1 and L2 be two intersecting lines and Q a point in the plane defined by L1 and L2. The image of Q under parallel projection on line L2 in the direction of L1 is obtained in the following ways

(a) Draw a line parallel to L1 through point Q as shown in the diagram below

(b) Let the line drawn meat L2 at point Q′.

(c) Q′ is the image of Q under projection on L2 in the direction of L1.

In a similar way, find the images of U,S,V,T and R under the same parallel projection.

Fig 8.9 shows the points U,S,Q,V,T,R and their corresponding images U′,S′,Q′,V′,T′,R′ under the parallel projection on L2 in the direction of L1.

Remember!

Fig 8.9 can only represent a parallel projection on L2 in the direction of L1 if and only if the line segment UU′, SS′, QQ′, VV′, TT′, and RR′ are all parallel to L1. How can you verify that this is so?

The images for the points U, S, Q, V, T, R under parallel projection are U′, V′, Q′, T′ and R′ as figure 8.9 shows

8.1.3 Parallel projection of a Line Segment on a line

On the diagram below, consider the line segment AB and the line L1 and L2 as figure 8.10 shows.

The line segment AB is projected by first projecting A on L2 in the direction of L1 to give the image A´ and then projecting B on L2 to give B´. This is shown in figure 8.1

The parallel projection of line AB segment   on L2 is line segment A´B´.

8.1.4 Properties of parallel projection

Activity 8.3

Refer back to Fig 8.11, find the lengths of:

1. (a) AB  

(b) A´B´ what do you notice?

2. (a) AA´

    (b) BB′ what do you notice?

3. Evaluate (a)  A´B´/AB
                  (b)   BB´/AA´
       4. Measure angles:

(a) <AA´B´     

(b) <BB´O What do you notice?

(c)  What can you say abut   the interior angles of   the figureAA´B´B?

Note that:

i) The parallel projection on one line, all images are formed on that line.

ii) A point on the line is mapped onto itself under parallel projection on the same line. Such a point is said to be invariant.

iii) Invariant points are those points which lie exactly on the line of projection under parallel projection. For example in Fig. 8.12, A, B, C and D are invariant points. In figure 8.12, the line segment AB is equal to CD i.e. AB =CD

The parallel projection of line segment AB is A´B´and the parallel projection of the segment CD is C´D´. If two line segments have the same length, then their parallel projection have the same length as well. If the segments are in the same direction.

iv) If a line segment, say AB  to be projected is parallel to the direction of the projection, then the two points have the same image. In fig 8.13, points C and D define a line segment CD which is parallel to L1, the line giving the direction of the parallel projection.

v) The image of midpoint of line segment is the midpoint of the image of the segment. (see Fig. 8.14)

8.1.5 Parallel projection of a geometric figure on a line

Activity 8.4

Given that L1 and L2 represent a line of parallel projection, and the direction of the projection respectively;

1. Draw the lines L1 and L2 to intersect at point O at 40°.

2. Draw a simple geometric figure i.e. a triangle, a rectangle, trapezium etc away from the line L1 and L2.

3. Construct the parallel projections of A, B and C on L1 in the direction of L2. Label the image of A,B and C.

4. Fig 8.12 shows the mapping of ∆ABC onto the projection line L1 parallel to line L2.

5. Describe the image of ABC.

6. In a similar way identify a line of projection and another one to show the direction of the projection.

7. Find the image of another plane shape under a parallel projection. 8. Comment on your findings.

Note

In Fig. 8.12, ∠ABC is the object.

• The three vertices have been projected on the line L1.

• A′B′ and C′ are the images of A,B,C respectively.

∴ ∠ABC is mapped onto a line segment
 A´C´

1. Fig. 8.16 represents a line segment  AB and a half line L intersecting at point B, at an angle of 30°

a) Given that AB = 12 cm and that  BC=  CD= DE = EF . Copy this figure accurately. To mark off C, D, E and F, use a pair of compasses, start from B using a radius of 3 cm

b) Join F to A.

c) Find the parallel projections of C, D and E on AB in the direction FA. Let C, D, E to be projections of C’, D’ and E’.

d) As accurately as possible, measure the lengths of the line segments BC′, C′D′, D′E′ and E′A, stating your measurements to the nearest mm. What do you notice?

ii) What do you notice?

f) What is the image of F under this projection? 

2. Using plain paper, ruler and compasses, a) Draw two lines L1 and L2 to intersect at a point O at an angle of 60°

b) Mark two points A and B as shown so that line segment  is not parallel to L1 or L2

c) Draw the projection of point A on L2 in the direction of L1 , then draw the projection of points B in the same way.

d) If A′and B′are the images of A and B respectively, describe the shape of the figure marked as AA′B′B.

e) Use your protractor to measure the interior angles of the figure. Name the two angles you could have stated without measuring why?

3. a) Draw a pair of parallel lines L1 and L2

b) Draw another line L3 to meet both L1 and L2 at a non-right angle at points A and B respectively.

c) Pick another point P on L1 and draw its parallel projection on L2 in the direction L3.

d) If the projection of P on L2 is denoted as Q, what can you say about figure PABQ? e) List the properties of the figure PABQ.

4. a) On a graph paper, draw lines whose equalities are (i) y + x = 5 (ii) y = 1/2 x

b) Let the lines meet at point P.

c) Find the parallel projection of a point A(-1, 3) on line y = 1/2 x in the direction y + x = 5. Let the image of A in this projection be denoted by A′.

d) Find the parallel projection of the point A on the line y + x = 5 in the direction y = 1/2 x, and let the image of A in this projection be denoted as A”.

e) State the coordinates of the points P, and A”. Name the geometric figure AA′PA′′and list its properties

5. Fig 8.19 shows object ∆ABC on the cartesian plane. Using line y = -x as the direction of parallel projection, find the coordinates of the image of ∆ABC on: (i) x-axis (ii) y-axis.

8.2 Orthogonal projection

8.2.1 Introduction to orthogonal projection

Activity 8.5

• Draw a line segment AB on a piece of paper. Above the line, mark point P as shown in Fig. 8.20.

• Place a pair of compasses at point P and mark two arcs on line AB as in Fig. 8.21.

• Without changing the radius of the compass, place the compass on each of the arcs already made on line AB. Make another pair of intersecting arcs below the line as shown in Fig. 8.18.

• Using a ruler, draw a line from P to meet AB at point P´as shown in Fig. 8.23.

From fig 8.19, PP´ is a perpendicular to AB and P´ is the image of P.

If a line segment PP´ meets another line say L2 at right angles at point P´, P´ is called orthogonal projection of point P in the line L2.

Orthogonal projection is the type of projection where the line of projection and the line giving the direction meet at 90°. For example, in the figure 8.23, L1 is the line of projection, L2 is the line giving the direction, and P is the object point.

Orthogonal projection can be regarded as a subject of parallel projections. In such a case it is not necessary to draw the first line if the line to show the direction of the projection is already known (see Fig. 8.25).

8.2.2 Orthogonal projection of a line segment on a line

Activity 8.6

Consider the horizontal line AB and a line PQ above line AB in Fig. 8.26.

1. Construct a perpendicular from point P to meet the line AB at 90°.

2. Put the compass at point P. Make two arcs on line AB.

3. Without changing the radius of the compass transfer it to the two arcs made on line AB and make other arcs that intersect below AB.

4. Using a ruler, draw the line from P to meet the point of intersection of the arcs below the line AB.

5. Repeat procedures 2-4 for point Q.

6. Discuss your drawing with other classmates.

P´Q´ is the orthogonal projection of PQ  on line AB

8.3.3 Orthogonal projection of a geometric figure on a line

Activity 8.7

1. Let L1 represent the line of projection and figure ABCD the object to be projected.

2. Construct the image points of ABCD using orthogonal projections on line L1 i.e. construct line segments from points A,B,C,D to the line of projection. On line L1, mark clearly the image points A´B´C´ and D´

3. Describe the shape of the image figure A´B´C´D´.

4. What properties of the trapezium ABCD have been preserved?

5. Repeat the activity using another geometric shape.

6. Comment on your findings.

Note:  For the construction in activity 8.7 there was no need to indicate the direction of the projection. Explain the reason

Properties of orthogonal projection:

• The projection meets the line of projection at 90°.

• Preserves ratios of corresponding line segments and ratio of corresponding projections.

• Preserves the distance between line segments and pairs of corresponding points.

Exercise 8.2

1. Given the points A, B, C and D which are on the line L2. If the line L2 meets L1 at the angle of 60º. Point M is
 4 cm from A. (See figure 8.28)

(a) Using a ruler and a protractor, draw accurately the diagram above. (b) Find the orthogonal projection of the points A, B, C and D on L1 (c) Find the ratios:

(d) What do you notice about the results from c(iv) and c(vii) above?

2. Study the graph below and answer the questions that follow.

The triangle ABC is given orthogonal projection on x axis.

(a) Find the coordinates of A', B' and C' under the projection on x-axis (b) Measure lengths:

(i) A'B'

(ii) B'C'

(iii) A'C'

3. Given that line AB=8 cm and it makes an angle of 35° to line L2 as shown in fig. 8.30.

By construction, find the length of orthogonal projection of line AB on line L2. 4. Given that BC´ =3.5 cm is orthogonal projection of BC=6 cm on line K as shown in Fig. 8.31

(i) By construction, determine angle CBC´.

(ii) Find the length BD if its orthogonal projection on line K is BD´ = 7 cm.

5. Use Fig 8.32 to find the projection of PQRS on the line y = x in the direction y = -x. 

(a) State the coordinates of P´Q´R´S´

(b) Find the projection of PQRS on the line y = -x in the direction  y = x (let the image points of PQRS be P´´Q´´R´´S´´

Unit summary

1. Parallel projection is projection in which the projection rays (lines) through the object are parallel to one another.

2. Properties of parallel projection

(i) The image of the midpoint of an object is the midpoint of the image (ii) The ratios of image to corresponding object lengths is constant      For example

(ii)  A point on the line or plane of projection is mapped onto itself by a parallel projection

(iv)  All the points on the object that is line segment parallel to the projection rays are mapped onto one image point.

3. Orthogonal projection is a special parallel projection in which the projection rays (lines) through the object are parallel to one another and perpendicular to the line or plane of projection.

4. Orthogonal projection has all the properties of parallel projection.

Unit 8 test

1. Given that line AB = 8 cm and it makes an angle of 45° to line L as shown in Fig. 8.34.

By construction, find the length of orthogonal projection of line AB on line L.

2. a)  Draw horizontal line AB = 8 cm
  and AC = 5 cm with AC   intersecting AB at 30º.

(b) Draw a diagram showing orthogonal projection of point C onto line AB.

(c)  Measure the length C to C´.

3.  Draw the image showing the orthogonal projection of line segment PQ onto the line AB as shown in the figure 8.35

4. A student moved from point A, 6 m due east to B and changed the direction 8 m due north to point C as shown in Fig. 8.36.

(a) Using a ruler construct accurately the diagram showing the course.

(b) Draw line AC.

(c) Find the length of the line AC.

(d) Locate the midpoint D of AC.

(e) Project point D orthogonally to AB.

(f) If the image of point D is formed on point E along AB, find the ratio of length DE:CB.