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Model Transformation and Coordinate System Transformation


Figure 5.26 Perspective projection with one projection center

2) Two-center perspective projection


Figure 5.27 Two-center perspective projection

[ Tc ] = [ Tpq ][ Tz ]


1 0 0 𝑝

1 0 0 0

1 0 0 𝑝

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= 0 1 0 π‘ž0 1 0 0 = 0 1 0 π‘ž


1 0 0 𝑝

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[T pq ]= 0 1 0 π‘ž


1 0 0 𝑝

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[xyz 1]= 0 1 0 π‘ž=[xyz (px+qy+1) ]

[x' y' z' 1]= π‘₯ 𝑦 𝑧 1

(𝑝π‘₯ +π‘žπ‘¦ +1) (𝑝π‘₯ +π‘žπ‘¦ +1) (𝑝π‘₯ +π‘žπ‘¦ +1)


Two projection centers: [ -1/p 0 0 1 ] and [ 0 -1/q 0 1 ]

The vanishing point (VP) on the x and y axes is respectively: [ 1/p 0 0 1 ] and [ 0 1/q 0 1 ].

3) Three-center perspective projection

[ T pqr ] = [T p ][T q ][T r ]


1 0 0 𝑝

1 0 0 0

1 0 0 0

1 0 0 𝑝

=0 1 0 00 1 0 π‘ž0 1 0 0=0 1 0 π‘ž

0 0 1 0

0 0 0 1

0 0 1 0

0 0 0 1

1 0 0 𝑝

0 0 1 π‘Ÿ

0 0 0 1

1 0 0 0

0 0 1 π‘Ÿ

0 0 0 1

1 0 0 𝑝


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π‘ž 0

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π‘ž


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[T c ]=[T pqr ].[T z ]= . = 0 0 1 π‘Ÿ 0 0 0 0 0


1 0 0 𝑝

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[xyz 1]= 0 1 0 π‘ž=[xyz (px+qy+1) ]


[x' y' z' 1]= π‘₯ 𝑦 𝑧 1

(𝑝π‘₯ +π‘žπ‘¦ +π‘Ÿπ‘§ +1) (𝑝π‘₯ +π‘žπ‘¦ +π‘Ÿπ‘§ +1) (𝑝π‘₯ +π‘žπ‘¦ +π‘Ÿπ‘§ +1)



Figure 5.28 Three-center perspective projection

Three projections:


on the x-axis at point [ -1/p 0 0 1 ], on the y-axis at point [ 0 -1/q 0 1 ]

on the z axis at point [ 0 0 -1/r 1 ] The vanishing point -VP will correspond to the values:

[ 1/p 0 0 1 ], [ 0 1/q 0 1 ], [ 0 0 1/r 1 ]

5.5. Model transformation and coordinate system transformation

Up to this point, we have considered three-dimensional transformations as the operation of moving a point (an object) from one position to another in a coordinate system. However, many times, we need to consider objects in different coordinate systems, and want to convert from one coordinate system to another. For example, in the process of displaying a three-dimensional object, we need to place an object in the coordinate system common to all objects in the scene (the real-world coordinate system), then, defining the view, we convert from the real-world coordinate system to the view coordinate system, and finally we must convert from the view coordinate system to the device coordinate system, where the objects will be displayed.

When modeling objects, we usually describe them in a local coordinate system, which is most convenient for modeling. Then, using transformations, we place them in the scene to be displayed. This approach allows us to model objects by type instead of by type. Converting object descriptions from one coordinate system to another follows a similar process to that in two-dimensional graphics. We need to build a transformation matrix to match the coordinate axes of the two systems. First, we need to perform a translation so that the two coordinate origins coincide. Then, we must perform a series of rotations to match the corresponding coordinate axes. If the coordinate systems use different measurement scales, we must perform another scale transformation to unify the coordinate systems.


Figure 5.27Coordinate modelling and transformation

If the second coordinate system has its origin at (x 0 , y 0 , z 0 ) and the basis vectors are as shown in Figure 6.15 (corresponding to the first coordinate system), we first need to perform a translation operation T( -x 0 ,-y 0 ,-z 0 ). Then we construct a rotation matrix R based on the basis vectors. This matrix will transform the unit vectors u' x , u' y , u' z into the x, y, z axes, respectively.


The matrix of the coordinate system transformation is the product TR. This matrix transforms one Cartesian coordinate system into another, whether they are right-handed or left-handed coordinate systems.


Figure 5.28 Coordinate system conversion

QUESTIONS AND EXERCISES CHAPTER 5

Choose one correct answer for the following questions.

1. In 3D, the vector coordinates of point P(-1,3,6) are:

[a]--(-2, 6, 12)

[b]--(-2, 6,12,2)

[c]--(- 0.5, 1.5, 3, 0.5)

[d]--(-a, 3a, 6a, a) where a β‰  0

2. In 3D there is a point Q(-4, 1.6, -7), symmetrical through the ox axis, Q' is: [a]--(-4, -1.6, -7)

[b]--(4 , -1.6 , 7)

[c]--(4 , -1.6 , -7)

[d]--(-4 , -1.6 , 7)

3. In 3D, there is a point M(5.34, - 31.8, - 0.3), symmetrical to the xOy plane, Q' is: [a]--(-5.34, 31.8, 0.3)

[b]--(5.34 , - 31.8 , 0.3)

[c]--(-5.34, - 31.8, - 0.3)

[d]--(5.34 , 31.8 , -0.3)

4. In 3D, there is a point Q(-4, 1.6, -7), symmetrical through the oz axis, Q' is: [a]--(4, -1.6, 7)

[b]--(4 , -1.6 , -7)

[c]--(-4 , 1.6 , 7)

[d]--(-4 , 1.6 , -7)

5. In 3D, there is a point A(2, -3, 1.4) after transforming it to be 2 times higher (according to oy), 1/2 thinner (according to oz) and the facade increased 3 times, then the obtained Q' is:

[a]--(6 , -9 , 0.7)

[b]--(4 , -1.5 , 4.2)

[c]--(1 , -6 , 4.2)

[d]--(6 , -6 , 0.7)

6. In 3D there is a point A(2.7, -3 , 2.5), rotate A around the oz axis at an angle of 90o. The new coordinates A' will be:

[a]--(2.7, -2.5 , -3 )

[b]--(3 , 2.7 , 2.5)

[c]--(2.5 , 3, 2.7 )

[d]--(-3 , 2.7 ,2.5)

7. In 3D there is a point B(-11.5, -2, 4.2), rotate B around the ox axis at an angle of -90o. The new coordinate B' will be:

[a]--(4.2 , -11.5, 2)

[b]--(-2 , 4.2 , -11.5)

[c]--(-11.5 , 4.2 , 2)

[d]--(11.5 , 4.2 , -2)

8. In 3D there is a point H(2, -12, -4), rotate H around the oz axis at an angle of 60o. The new coordinates H' are:

[a]--(2; -6 - 2√3; - 6√3 + 2 )

[b]--(2; -6 +2√3; - 6√3 -2 )

[c]--(1+ 6√3; √3 -6 ; 4)

[d]--(1- 6√3; √3 -6 ; 2)

9. In 3D there is a point H(2,-4,6), rotate point H around the oy axis at an angle of 45o then take symmetry through the xOy plane. Point H' has coordinates:

[a]--(-4√2; -4; 4√2)

[b]--(-4√2; 4; 4√2)

[c]--(4√2; -4; -4√2)

[d]--(4√2; 4; -4√2)

10. In 3D, there is a point V(2,-4,6). Rotate point V around the ox axis at an angle of -45o and then reflect it through the xOy plane. Point V' has coordinates:

[a]--(2; 3√2; 5√2)

[b]--(2; - 5√2; -√2)

[c]--(2; -3√2; - 5√2)

[d]--(2; 5√2; -√2)

11. In 3D there is a point L(2,-4,6), rotate point L around the oz axis by 30o then double the zoom. Point L' has coordinates:

[a]--(√3 - 2 ;-4√3; 12)

[b]--(2√3 - 2 ; -1-4√3; 12)

[c]--(√3 - 2 ; 2 -√3; 6)

[d]--(2√3 + 2 ; 1+ 4√3; 12)

12. Given diamond ABCD with coordinates A(5,6,1), B(0,0,0), C(3,2,5) and D(8,2,4). Rotate the diamond around the oy axis by an angle of 900. The new coordinates of the diamond are:

[a]--A'(1,6,-5), B'(1,2,2), C'(5,2,-3) and D'(2,-4,8)

[b]--A'(1,6,-5), B'(0,0,0), C'(2,5,-3) and D'(2,4,-8)

[c]--A'(-1,5,-1), B'(0,0,0), C'(2,3,5) and D'(2,4,8)

[d]--A'(1,6,-5), B'(0,0,0), C'(5,2,-3) and D'(4,2,-8)

13. "The method of …… is not a projection" . Choose one option to fill in the blank. [a]--Axis projection

[b]--Isometric projection [c]--Cabinet projection [d]--Mapping projection

14. Which of the following projections is not an orthographic projection? [a]--Vertical projection

[b]--Sectional projection [c]--Plane projection [d]--Edge projection

15. The two commonly used angles in oblique projection are the angles with: [a]--tg Ο† =1 and tg Ο† =2

[b]--Sinφ =1 and Cosφ =-1 [c]--Cos φ =1 and tg φ = 1 [d]--Sinφ =1 and Sinφ = -1

16. When the projection angle is chosen so that tg Ο† = 2, the resulting scene is called a projection: [a]--Same-size projection

[b]--Orthogonal projection of the geometric axis [c]--Cabinet

[d]β€”Cavalier

17. ―Trimetric projection is a parallel projection with projection rays perpendicular to the projection screen, the projection is obtained after rotating the object so that 3 sides of the object are visible (usually the projection plane is z=0) and the aspect ratio .... ―. Which option to fill in the blank is correct?

[a]--fx = fy = fz = 1/2 [b]--fx β‰  fy β‰  fz

[c]--fx = fy β‰  fz

[d]--fx = fy

18. ―Isometric projection is a parallel projection with projection rays perpendicular to the projection screen, the projection image is obtained after rotating the object so that 3 sides of the object are visible (usually the projection plane is z=0) and the aspect ratio .... ―. Which option to fill in the blank is correct?

[a]--fx = fy = fz = 1/2 [b]--fx β‰  fy β‰  fz

[c]--fx = fy


[d]--fx = fy = fz = (2/3)

19. The Cabinet projection is a projection with an aspect ratio of: [a]--f=0.8165

[b]--f=1.2

[c]--f=1/2

[d]--f=1

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