LAB 3: COMPUTER GRAPHICS AND POLYGONS

 

Mathematics:

 

When a three-dimensional object is projected onto a two-dimensional computer screen, many aspects of the image may be lost for the user's eyes. For example, we will be working with an object that is displayed by a finite number of straight line segments, such as the truncated pyramid with hexagonal base. If the vertical segments of the pyramid located at different three-dimensional faces merge and collapse in the two-dimensional screen, then an user may get a wrong impression about the original image.

The three-dimenional object is defined by a number of vertices:

(x1,y1,z1), (x2,y2,z2), , (xn,yn,zn),

and a number of pairwise line segments connecting the vertices:

(k1k2), (k2k3), , (km-1km).

 

 

The truncated pyramid with hexagonal base shown above is defined by n = 12 vertices:

P1(1,-0.8,0)

P2(0.5,-0.8,-0.866)

P3(-0.5,-0.8,-0.866)

P4(-1,-0.8,0)

P5(-0.5,-0.8,0.866)

P6(0.5,-0.8,0.866)

P7(0.84,-0.4,0)

P8(0.315,0.125,-0.546)

P9(-0.21,0.65,-0.364)

P10(-0.36,0.8,0)

P11(-0.210,0.65,0.364)

P12(0.315,0.125,0.546)

 

 

and by m = 18 line segments between the vertices:

(P1P2)

(P2P3)

(P3P4)

(P4P5)

(P5P6)

(P6P1)

(P7P8)

(P8P9)

(P9P10)

(P10P11)

(P11P12)

(P12P7)

(P1P7)

(P2P8)

(P3P9)

(P4P10)

(P5P11)

(P6P12)

 

 

The two-dimensional screen projection is defined at the plane (x,y) with z = 0. After the screen projection, the image of the truncated pyramid with hexagonal base transforms to a set of three polygons shown here:

At the plane (x,y) with z = 0, 12 vertices of the pyramid collapse into 8 vertices of the image and 18 line segments of the pyramid collapse into 10 edges of the three polygons. A better view of the pyramid on the screen can be achieved by applying different linear transformations to the vertices (xk,yk,zk), such as scaling, translation, and rotation. For example, the picture below represents the truncated pyramid relatively well:

Objectives:

         develop three MATLAB functions performing scalings, translations, and rotations at a given set of vertices of a three-dimensional object

         understand effects of scalings, translations, and rotations applied to vertices of a three-dimensional object

         develop a MATLAB function drawing a two-dimensional image of a three-dimensional object on the computer screen.

         exploit linear transformations to find the best view of a given three-dimensional object on the computer screen.

 

MATLAB functions performing scalings, translations, and rotations at a given set of vertices

 

Let us define a 3-by-n matrix P for a given set of vertices of a three-dimensional object:

P =

This matrix is called the coordinate matrix of the view. The linear transformations such as scalings, translations, and rotations are applied to the matrix P.

 

A 3-by-3 scaling matrix S is defined as

S =

The scaling matrix S transforms a unit cube (1,1,1) into a scaled cube (,,). The multiplication matrix P' = S*P transforms vertices of the original picture into vertices of new picture. Scaling in z is not visible after the projection of the object onto the plane at z=0.

 

A 3-by-n translation matrix T is defined as

T =

The sum matrix T + P moves each vertex (xk,yk,zk) of the original picture into the vertex (xk+x0,yk+y0,zk+z0), translating the whole picture in the direction of (x0,y0,z0). Translation in the z-direction is not visible after the projection of the object onto the plane at z=0.

 

3-by-3 rotation matrices Rx,Ry,Rz are defined as

Rx = , Ry = , Rz =

The matrix Rz rotates the (x,y) plane about the z-axis at the angle counter-clockwise. The matrices Rx and Ry perform rotations in the (y,z) and (z,x) planes, respectively. The rotation matrices are orthogonal matrices, i.e. det(Rz) = 1 and RzT = Rz-1. The multiplication matrix Rx*Ry*Rz*P rotates all vertices of the original picture onto the same angles in the (x,y,z) space.

 

Steps in writing the MATLAB functions:
  1. Define the MATLAB function "Scale" with input parameters [P,,,] and output parameter [P'].
  2. Write a short commented description of the function.
  3. Construct the scaling matrix S and compute P'.
  4. Define the MATLAB function "Translate" with input parameters [P,x0,y0,z0] and output parameter [P']
  5. Write a short commented description of the function.
  6. Construct the translation matrix T and compute P'.
  7. Define the MATLAB function "Rotate" with input parameters [P,, index] and output parameter [P'].
  8. Write a short commented description of the function.
  9. Construct the rotation matrix R depending on the index and compute P'.

MATLAB function drawing a two-dimensional image of a three-dimensional object on the computer screen

 
Let us define a 2-by-m matrix C for a given set of line segments between the vertices of the three-dimensional object:

C =

It is recommended to use an advanced MATLAB data structure "struct" and represent C as an array of structures. The three-dimensional object can be displayed at the computer screen if P and C are given. The two-dimensional object is displayed as a set of polygons, where the line segments are connected the vertices at the plane (x,y), i.e. the object is projected into the plane z = 0.

 

 

Steps in writing the MATLAB function:
  1. Define the MATLAB function "Draw" with input parameters [P,C] and no output parameters.
  2. Project P into a 2-by-n matrix P2, at the plane z = 0.
  3. Loop through columns of C and draw line segments between the corresponding vertices of the picture.
  4. Scale axis, fill the polygons, give a title and labels to the figure.

 

Exploiting the MATLAB functions:

  1. Define P and C for the truncated pyramid with hexagonal base.
  2. Draw the pyramid on the screen by calling the function "Draw".
  3. Redraw the pyramid after the scaling transformation with alpha = 1.8, beta = 0.5, and gamma = 3.
  4. Redraw the original pyramid after the translation with x0 = 1.2, y0 = 0.4, and z0 = 1.7.
  5. Redraw the original pyramid after rotation about the x-axis with theta = 900.
  6. Redraw the original pyramid after rotation about the y-axis with theta = 900.
  7. Redraw the original pyramid after rotation about the z-axis with theta = 900
  8. Redraw the original pyramid after composite rotations: about the x-axis with theta = 300, about the y-axis with theta = -700, about the z-axis with theta = -270.

 

 

QUIZ:

 

  1. Draw the truncated pyramid after the following five transformations:

a)      Scale the image by a factor of 0.5 in the x-direction, 2 in the y-direction, and 1/3 in the z-direction.

b)      Translate 0.5 unit in the x-direction

c)      Rotate 200 about the x-axis

d)      Rotate 450 about the y-axis

e)      Rotate 900 about the z-axis.

 

  1. Draw the truncated pyramid after the following seven transformations:

a)      Scale the image by a factor of 0.3 in the x-direction and 0.5 in the y-direction.

b)      Rotate 450 about the x-axis

c)      Translate 1 unit in the x-direction.

d)      Rotate 350 about the y-axis

e)      Rotate 450 the z-axis.

f)       Translate 1 unit in the z-direction.

g)      Scale by a factor of 2 in the x-direction.