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Surfaces

Human Centered 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

 Equation

Bezier surface – cont’

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, (v) is defined by t₀,t₁,…,t _+m

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



   











 

n

0 i

m

0

j 1 m 1

1 n 1

k j,

k i, j

i, t v t

s u s

(v) (u)N

N v)

(u,

l

P l

P

l

l l

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 , knot t_p (p=0,1,…,m+ )

 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

Interpolation surface

 Generate a surface from data points

 Essential in reverse engineering

 Derive B-spline surface passing through data points Q_p,q(p=0,…n, q=0,…m).

 Constraints: (n+1)×(m+1)

 B-spline’s control points:

minimum (n+1)×(m+1) control points are needed.

Interpolation surface – cont’

 P_i,j should be obtained

 Let u_p, v_q to be parameter value of Q_p,q

 u_p is obtained while interpolating Q_0,q, Q_1,q,…, Q_n,q

 v_q is obtained while interpolating Q_p,0, Q_p,1,… , Q_p,m



 



n

0 i

m

0 j

j, k

i, j

i, N (u)N (v)

v)

(u, P ^l

P



 











n m

n

0 i

m

0 j

q j, p k i, j i, q

p, P N (u )N (v )

Q ^l

Interpolation surface – cont’

 C_i(v₀) (i=0,…,n) are control points of B-spline interpolating Q_0,0, Q_1,0, …, Q_n,0

 Similarly C_i(v₁) (i=0,…,n) are control points of B-spline interpolating Q_0,1, Q_1,1, _Q2,1, … , Q_n,1

























n

0 i

p k i, m i m

p,

n

0 i

p k i, 1 i p,1

n

0 i

p k i, 0 i p,0

n

0 i

p k i, q i q

p,

) (u )N (v

) (u )N (v

) (u )N (v

) ( 0

) ( )

(u )N (v

C Q

C Q

C Q

C Q



a in q for m to substitute

a

Interpolation surface – cont’

 Derive P_i,j from C_i(v_q) (q=0,…,m)

 Substitute 0 to m for q in (b)







m

0 j

q q

i(v ) P^i,jN^j,l(v ) (b)

C

Interpolation surface – cont’

 P_i,jare control points of B-spline interpolating C_i(v₀), C_i(v₁),…,C_i(v_m)

 P_0,j are from C₀(v₀), C₀(v₁), …, C₀(v_m)

 P_1,j are from C₁(v₀), C₁(v₁), …, C₁(v_m)

 P_n,j are from C_n(v₀), C_n(v₁), …, C_n(v_m)

Interpolation surface – cont’

Interpolation surface – cont’

Interpolation surface – cont’

 Order k, are usually set to be 4.

 Knot values

 (m+1) set of knot values are obtained during interpolation along u direction

 (m+1) set are averaged to result knot values in u direction

 Knot values in v direction are derived in the same way



Intersection between surfaces

 Many points on the intersection curves are obtained by numerical solution

 Intersection curve should be calculated in Boolean operation, surface modeling, etc.

 P(u,v) – Q(s,t) = 0

 Four unknowns u, v, s, t with three scalar eqs.

 Set one parameter value and determine remaining parameter values, repeat with many values

 Dependent on Initial value.

 Cannot guarantee all intersection curves

Subdivision method

 Subdivide both surfaces until control polyhedron approximates the surface

 Look for intersecting pairs of planar quadrilaterals and calculate the intersection

 Initial values are obtained from the intersection of planar quadrilaterals

Subdivision method – cont’

 Start from one end of intersection and look for the next pair to be considered and calculate intersection

 Until surface boundary is reached

 Traverse in the opposite direction starting from the other end

 Start from different pair until all the intersection curves are obtained

Tracing intersection segments