定义：
1.旋转：绕原点旋转，逆时针旋转为正，旋转角度为[ heta ]
2.平移：平移向量为[({x_0},{y_0})]
旋转矩阵：
1.点旋转：
[{P_{rot}} = left[ {egin{array}{*{20}{c}}
{cos heta }&{-sin heta }&0\
{sin heta }&{cos heta }&0\
0&0&1
end{array}}
ight]P]
[P = left[ {egin{array}{*{20}{c}}
x\
y\
1
end{array}}
ight]]
2.曲线C旋转：
曲线C为：
[A{x^2} + Bxy + C{y^2} + Dx + Ey + F = 0]
[{left[ {egin{array}{*{20}{c}}
x\
y\
1
end{array}}
ight]^T}left[ {egin{array}{*{20}{c}}
A&{B{
m{/2}}}&{D{
m{/}}2}\
{B{
m{/2}}}&C&{E{
m{/2}}}\
{D{
m{/}}2}&{E{
m{/2}}}&F
end{array}}
ight]left[ {egin{array}{*{20}{c}}
x\
y\
1
end{array}}
ight]{
m{ = 0}}]
[C = left[ {egin{array}{*{20}{c}}
A&{B{
m{/2}}}&{D{
m{/}}2}\
{B{
m{/2}}}&C&{E{
m{/2}}}\
{D{
m{/}}2}&{E{
m{/2}}}&F
end{array}}
ight]]
[egin{array}{l}
{C_{Rot}} = {R^T}CR\
R = left[ {egin{array}{*{20}{c}}
{cos heta }&{sin heta }&0\
{ - sin heta }&{cos heta }&0\
0&0&1
end{array}}
ight]
end{array}]
平移
[egin{array}{l}
{C_{Trans}} = {T^T}CT\
T = left[ {egin{array}{*{20}{c}}
1&0&{ - {x_0}}\
0&1&{ - {y_0}}\
0&0&1
end{array}}
ight]
end{array}]
实验验证
1.点的旋转（红色是旋转后的）
theta = pi/2;
Rot1 = [cos(theta) -sin(theta) 0;
sin(theta) cos(theta) 0;
0 0 1;]
P = [1,0,1]';
P1 = Rot1 * P;
figure
plot(P(1),P(2),'bx','linewidth',15)
hold on;
plot(P1(1),P1(2),'ro','linewidth',15)
xlim([-0.5,2])
ylim([-0.5,2])
grid on;
旋转角度90
旋转角度45
2.曲线的旋转和平移（红色是旋转平移后的）
a1 = 3;
b1 = 2;
C1 = [1/a1.^2 0 0;
0 1/b1.^2 0;
0 0 -1;];
figure(1)
syms x y
f0=ezplot( C1(1,1)*x^2+ C1(2,2)*y^2 +C1(3,3) + 2*C1(1,2)*x*y + 2*C1(1,3)*x +2*C1(2,3)*y,[-6,6],[-6,6]);
set(f0,'Color','b','LineWidth',1.5)
hold on;
grid on;
% 画椭圆1
% 画椭圆1旋转平移
theta = pi/4;
Rot = [cos(theta) sin(theta) 0;
-sin(theta) cos(theta) 0;
0 0 1;]
x0 = 1;
y0 = 2;
T = [1 0 -x0;
0 1 -y0;
0 0 1;];
C1 = T'*Rot'*C1*Rot*T
syms x y
f1 = ezplot( C1(1,1)*x^2+ C1(2,2)*y^2 +C1(3,3) + 2*C1(1,2)*x*y + 2*C1(1,3)*x +2*C1(2,3)*y,[-6,6],[-6,6]);
set(f1,'Color','r','LineWidth',1.5)
