Structure-deforming(變形) transformations--3D(1)

(x,y,z) undeformed vertex

(X,Y,Z) deformed vertex

X=Fx(x)

Y=Fy(y)

Z=Fz(z)

X=Sx(x)

Y=Sy(y)

Z=Sz(z)

X=xcosθ-γsinθ

Y=xsinθ+γcosθ

Z=z

θ=f5(z)

f'(z) specifies the rate of twist per unit length along the z axis

公式

X=x

Y={-sinθ(z-1/k)+γ0 γmin<=γ<=γmax

{-sinθ(z-1/k)+γ0+cosθ(γ-γmin) γ<γmin

{-sinθ(z-1/k)+γ0+cosθ(γ-γmax) γ>γmax

Z={-cosθ(z-1/k)+1/k+γ0 γmin<=γ<=γmax

{-cosθ(z-1/k)+1/k+γ0+sinθ(γ-γmin) γ<γmin

{-cosθ(z-1/k)+1/k+γ0+sinθ(γ-γmax) γ>γmax

velocity速度 V=(v1,v2,v3)

Addition of Vectors

X=T+W

=(x1,x2,x3)

=(v1+w1,v2+w2,v3+w3)

Length of Vectors

|V|=(v1^2+v2^2+v3^2)^1/2

U=v/|v| normalized V

v=|v|u

cross product of V and W

X=VхW

=(v2w3-v3w2)i+(v3w1-v1w3)j+(v1w2-v2w1)k

i=(1,0,0)

j=(0,1,0)

k=(0,0,1)

Q(u,v)為平面Q 任一點(u,v)

Ns=∂Q/∂u + ∂Q/∂v

Np=V1x v2

Normal vector(法向量) and dot product

X=V∙W

=v1w1+v2w2+v3w3

cosθ=v∙w/|v||w|

|X|=|W|cosθ

=|W|∙V∙W/|V||W|

=V∙W

cosθ = v∙w/|v||w| 兩向量夾角

L入射光 R反射光

R=R1+R2

R1 = -L+R2

R=2R2-L

R2=(N∙L)N

R=2(N∙L)N – L