CS465 Fall 2006
Homework 4 Solution
-------------------

1.

Stem: Scale by (0.5, 8), then translate by 8 in y:

[1 0 0; 0 1 8; 0 0 1] * [0.5 0 0; 0 8 0; 0 0 1]
= [0.5 0 0; 0 8 8; 0 0 1]

Center: Just translate by 18 in y:

[1 0 0; 0 1 18; 0 0 1]

Petal 1: scale by (1,2), then translate up by 21.5:

[1 0 0; 0 1 21.5; 0 0 1] * [1 0 0; 0 2 0; 0 0 1]
= [1 0 0; 0 2 21.5; 0 0 1]

Leaf: scale by (1,4), translate by (0,4), then rotate by 60 degrees:

[cos(pi/3) -sin(pi/3) 0; sin(pi/3) cos(pi/3) 0; 0 0 1] * [1 0 0; 0 1 4; 0 0 1]
  * [1 0 0; 0 4 0; 0 0 1]
= [cos(pi/3) -4*sin(pi/3) -4*sin(pi/3); sin(pi/3) 4*cos(pi/3) 4*cos(pi/3); 0 0 1]
~= [0.5 -3.4641 -3.4641; 0.8660 2 2; 0 0 1]

Petal 2: scale by (1,2), then translate up by 1.5.  This places the petal in a
coordinate system where the center of the flower is at the origin.  Rotate by
2pi/5, then translate by 18 to put the center of the flower where it belongs:

[1 0 0; 0 1 18; 0 0 1]
  * [cos(2*pi/5) -sin(2*pi/5) 0; sin(2*pi/5) cos(2*pi/5) 0; 0 0 1]
  * [1 0 0; 0 1 3.5; 0 0 1] * [1 0 0; 0 2 0; 0 0 1]
= [cos(2*pi/5) -2*sin(2*pi/5) -3.5*sin(2*pi/5); sin(2*pi/5) 2*cos(2*pi/5) 3.5*cos(2*pi/5)+18; 0 0 1]
~= [0.3090 -1.9021 -3.3287; 0.9511 0.6180 19.0816; 0 0 1]


2.a.

Stem:
Scale along x axis by 0.5 and y axis by 8.  Translate along y axis by 8.

Center:
Translate along y axis by 18.

Petal 1:
Scale along y axis by 2.  Translate along y axis by 21.5.

Leaf:
Scale along y axis by 4.  Translate along y axis by 4.  Rotate about origin by pi/3.


2.b. Scale factors can be determined from the size of the ellipse relative to
the unit circle.  If we scale around a point c, then a point y on the axis
that has scale factor s will be mapped to a point y' on the axis that is s
times as far away from c.  That is, y' - c = s (y - c).  We know s, y, and y',
so we can solve for c:  c = (y' - sy) / (1 - s)

Petal 1:

Scale is by (1,2) and want to map (0,0) to (0,21.5).  Use s = 2, y = 0, y' =
21.5 to get c = -21.5.

So the answer is: Scale along x axis by 1 and along y axis by 2 with respect
to point (0, -21.5).

Leaf:

Let u = (cos(pi/3), sin(pi/3)) and v = (-sin(pi/3), cos(pi/3)).  Make a
coordinate system based on u and v with origin at (0, 0) in the xy coordinate
system.  In the u-v coordinate system, we have a scale of (1,4) and want to
map the point (0, -1) to (0, 0).  So use s = 4, y = -1, y' = 0 to get c =
-4/3.

So the answer (in u-v) is: Scale along u axis by 1 and along v axis by 4
with respect to point (0, -4/3).

And the answer in canonical coordinates is: Scale along the axis (cos(pi/3),
sin(pi/3)) by 1 and along the axis (-sin(pi/3), cos(pi/3)) by 4, around the
center (2/sqrt(3), -2/3).


3.
A. Scale along x axis by 0.5 and y axis by 8.  Translate along y axis by 8.
B. Translate along y axis by 18.
C. Scale along y axis by 2.  Translate along y axis by 21.5.
D. Scale along y axis by 4.  Translate along y axis by 4.  Rotate about origin by pi/3.
E. Rotate Petal 1 about the center of Center by 2*pi/5.
F. Rotate Petal 1 about the center of Center by 4*pi/5.
G. Reflect Left Half across y axis (the line x = 0).



And finally, for your edification here is the MATLAB function that generated
the picture in the handout:

function hw4

figure; grid on; hold on;
axis equal;

theta = linspace(0,2*pi,361);
unit_circle = [cos(theta); sin(theta); repmat(1, size(theta))];
plot(unit_circle(1,:), unit_circle(2,:));

A = [1 0 0; 0 1 8; 0 0 1] * [0.5 0 0; 0 8 0; 0 0 1];
stem = A*unit_circle;
plot(stem(1,:), stem(2,:));

B = [1 0 0; 0 1 18; 0 0 1];
center = B*unit_circle;
plot(center(1,:), center(2,:));

C = [1 0 0; 0 1 21.5; 0 0 1] * [1 0 0; 0 2 0; 0 0 1];
petal1 = C*unit_circle;
plot(petal1(1,:), petal1(2,:));

beta = 60*pi/180;
D = [cos(beta) -sin(beta) 0; sin(beta) cos(beta) 0; 0 0 1] * [1 0 0; 0 1 4; 0 0 1] * [1 0 0; 0 4 0; 0 0 1];
leaf = D*unit_circle;
plot(leaf(1,:), leaf(2,:));

beta = 360/5*pi/180;
E = [1 0 0; 0 1 18; 0 0 1] * [cos(beta) -sin(beta) 0; sin(beta) cos(beta) 0; 0 0 1] * [1 0 0; 0 1 -18; 0 0 1];
petal2 = E*petal1;
plot(petal2(1,:), petal2(2,:));

beta = 2*360/5*pi/180;
F = [1 0 0; 0 1 18; 0 0 1] * [cos(beta) -sin(beta) 0; sin(beta) cos(beta) 0; 0 0 1] * [1 0 0; 0 1 -18; 0 0 1];
petal3 = F*petal1;
plot(petal3(1,:), petal3(2,:));

G = [-1 0 0; 0 1 0; 0 0 1];
petal4 = G*petal3;
plot(petal4(1,:), petal4(2,:));
petal5 = G*petal2;
plot(petal5(1,:), petal5(2,:));
leaf2 = G*leaf;
plot(leaf2(1,:), leaf2(2,:));