Surfaces

Surfaces

CAD Lab.

Surfaces

 Parametric eq.

 Implicit eq

 Explicit eq

0 R

z y

x²  ²  ²  ² 

2 2

2 x y

R

z    

) 2 / 2

/

, 2 (0

sin cos

sin cos

cos )

, (





   











v u

v R

v u

R v

u R

v

u i j k

P

Bezier surface

      

 ⁿ

0 i

m 0 j

m j, n

i, j

i, B (u)B (v) 0 u 1, 0 1

v)

(u, P v

P

   

 ⁿ

0 i

n i, m

m, m i, m

1, i,1 m

0,

i,0B (v) B (v) B (v))B (u)

(P P  P

Bezier curve

Bezier surface – cont’

 Surface obtained by blending (n+1) Bezier curves

 (or by blending (m+1) Bezier curves)

 Four corner points on control polyhedron lie on surface

Bezier surface equation

P(0,0) P^{i , j}B (0)B^{i , n j, m}(0)

j 0 m

i 0

 n









 

  



 



 ⁿ 

0 i

n i,

P m

0 j

m j, j

i, B (0) B (0)

i,0



 



 

 P

 



 ^{Pi,0 i,n P0,0} i 0

n

B (0)

Bezier surface – cont’

 Boundary curves are Bezier curves defined by associated control points

 Bezier curve defined by P_0,0, P_0,1, …, P_0,m



 



 





 



 





 





 

 





 



 





m

0 j

m j, j 0,

m j, m

0 j

P n

0 i

n i, j i, n

0 i

m

0 j

m j, n

i, j i,

(v) B

(v) B

B

(v) (0)B

B v)

(0,

j 0 ,

0 u

P P P P

Bezier surface – cont’

 When two Bezier surfaces are connected, control points before and after connection should form straight lines to guarantee G1 continuity

B-spline surface

 N_i,k(u) is defined by s₀, s₁, …,s_n+k

 N_j,l(v) is defined by t₀,t₁,…,t_l+m

 If k=(n+1), l=m+1 and non-periodic knots are used, the resulting surface will become Bezier surface

    











 ⁿ 

0 i

m 0

j l 1 m 1

1 n 1

k l

j, k

i, j

i, t v t

s u s

(v) (v)N

N v)

(u, P

P

B-spline surface – cont’

 Bezier surface is a special case of B-spline surface.

 Boundary curves are B-spline curves defined by associated control points.

 Four corner points of control polyhedron lie one the surface (when non-periodic knots are used)

NURBS surface

 If h_i,j = 1, B-spline surface is obtained

 Represent quadric surface(cylindrical, conical, spherical, paraboloidal, hyperboloidal) exactly

1 m 1

1 n 1

k

0 0 0 0

t v

t

s u

s (v)

(u)

(v) (u)

v) (u,









 

 







 

 

 

j,l i,k

n i

m j

i,j n

i

m j

j,l i,k

i,j i,j

N N

h

N N

h P

P

NURBS surface – cont’

 Represent a surface obtained by sweeping a curve

NURBS surface – cont’

 Assume that v-direction of surface is a given P_j

 v-direction knot & order is the same as the NURBS Curve’s (order: l, knot: t_p)

 u-direction order is 2

 control point: 2

 u direction knot: 0 0 1 1

 P_0,j = P_j

 P_1,j = P_j + d a

 h_0,j = h_1,j = h_j from the given curve

Ex) Translate half circle to make cylinder

Ex) Translate half circle to make cylinder

 P₀=(1, 0, 0) h₀=1 P₁=(1, 1, 0) h₁=1/√2

 P₂=(0, 1, 0) h₂=1 P₃=(-1, 1, 0) h₃= 1/√2

 P₄=(-1, 0, 0) h₄=1

 P_0,0 = P₀, P_1,0 = P0+Hk h_0,0 = h_1,0=1

 P_0,1 = P₁, P_1,1 = P1+Hk h_0,1 = h_1,1= 1/√2

 P_0,2 = P₂, P_1,2 = P₂+Hk h_0,2 = h_1,2=1

 P_0,3 = P₃, P_1,3 = P₃+Hk h_0,3 = h_1,3= 1/√2

 P_0,4 = P₄, P_1,4 = P₄+Hk h_0,4 = h_1,4=1

 Knots for v: 0 0 0 1 1 2 2 2

Ex) Surface obtained by revolution

 Curve

 order l, knot t_p (p=0,1,…,m+l)

 control points P_j, h_j (j=0,1,…,m)

 Original control points needs to be split into 9.

Ex) Surface obtained by revolution – cont’

 P_0,j = P_j h_0,j = h_j

 P_1,j = P_0,j +x_j j = P_j + x_j j h_1,j = h_j·1/√2

 P_2,j = P_1,j - x_j i = P_j - x_j (i-j) h_2,j = h_j

 P_3,j = P_2,j - x_j i = P_j - x_j (2 i-j) h_3,j = h_j·1/√2

 P_4,j = P_3,j - x_j j = P_j - 2x_j i h_4,j = h_j

 P_5,j = P_4,j - x_j j = P_j - x_j (2 i+j) h_5,j = h_j·1/√2

 P_6,j = P_5,j + x_j i = P_j - x_j (i+j) h_6,j = h_j

 P_7,j = P_6,j + x_j i = P_j - x_j j h_7,j = h_j·1/√2

 P_8,j = P_0,j = P_j h_8,j = h_j

 u-direction order: 3