Determine a single 4 x 4 matrix that transforms a circle centered at the origin

Question

Determine a single 4 x 4 matrix that transforms a circle centered at the origin with radius r = 1/2 to the ellipse shown in Figure 1.

The ellipse is centered at the point (2; 1; 0), has major axis a = 1 and

minor axis b = 1

2 and is oriented at 45 degrees relative to the X-axis.

PROBLEM 2. Rotate the cube by = ô€€€45 about the line passing

through the point PO = (5; 0; 0) and having a unit vector ^ = (0; 0; 1).

Provide the 3D coordinates of all eight vertices in the original and

rotated locations.

PROBLEM 3. Determine the isometric projection of the cube using the

transformation. That is, determine the projected co-

ordinates (xv; yv) of all eight vertices in the viewing coordinate system

(VCS). Provide a plot of the projected cube in the XY-plane of the

VCS making sure to label the coordinate axes and vertices.

PROBLEM 4. Determine the orthographic projection of the rotated cube

of Problem 2 on the XZ-plane (TOP view). That is, determine the pro-

jected coordinates (xv; yv) of all eight vertices in the viewing coordinate

system (VCS). Provide a plot of the projected cube in the XY-plane

of the VCS making sure to label the coordinate axes and vertices.

PROBLEM 5.

Determine the perspective projection of the rotated cube

of Problem 2 using the transformation. That is,

determine the projected coordinates (xv; yv) of all eight vertices in the

viewing coordinate system (VCS) if the center of projection (COP) is

at the point C = (0; 0; 5). Provide a plot of the rotated and projected

cube in the XY-plane of the VCS making sure to label the coordinate

axes and vertices.

Explanation / Answer

use concept of transformation.....and to transform system from (x,y) to (u,v) use the concept of transformation .....and file JACOBIAN using d(u,v)/d(x,y).....d is partial differentiation...thanx ....do rate 5 stars

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