# Free- Free-form Surface I

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```					Free-form Surface I
Applications of Complex Surfaces
Surface Patch
A surface patch ⎯ a curved bounded collection of
points whose coordinates are given by continuous,
two-parameter, single-valued mathematical
expression.

Function of the form:
p (u, w ) = [x (u, w ) y (u, w ) z (u, w )]
T
-
-                          p(u,1) w=1
p(0,1)
-
p(1,1)
u=0             -
n(ui,wj)      u=ui
-                                  w=wj
p(0,w)
-                   u=1
p(ui,wj)
-
p=(1,w)

-                            -
p(0,0)                       p(u,0) w=0                     -
p(1,0)

z                 y

x
Type of Surface

•   Planar Surface
•   Bilinear Surface
•   Ruled (Lofted) Surface
•   Bi-cubic Surface
•   Bezier Surface
•   B-spline Surface
Planar Surface
- defined by three points and vectors

(        ) (
p ( u , w ) = p0 + u p1 − p0 + w p2 − p0      )        0 ≤ u ≤ 1; 0 ≤ w ≤ 1

^
s        ^
r
-        ^
n
p2                   -
p1
z         -
p0

y
x
Planar Surface
(         ) (
p ( u , w ) = p0 + u p1 − p0 + w p2 − p0   )    0 ≤ u ≤ 1; 0 ≤ w ≤ 1

^               ^
p ( u , w ) = p0 + u p1 − p0 r + w p2 − p0 s   0 ≤ u ≤ 1; 0 ≤ w ≤ 1

^    ^   ^
n = r× s — surface normal
^     p1 − p0       ^    p 2 − p0
r=              ; s=                    Normalized
p1 − p0            p 2 − p0    Direction Vectors
Planar Surface
^                  ^
p ( u , w ) = p0 + u p1 − p0 r + w p2 − p0 s                 0 ≤ u ≤ 1; 0 ≤ w ≤ 1

Ax + By + Cz + D = 0
n = Ai + Bˆ + Ck
ˆ     ˆ   j    ˆ
^     ^    ^                                       ^
s
n = r× s                                                     ^
r
-        ^
n
p2                   -
( P − P0 ) ⋅ n = 0
ˆ                                                                    p1
-
( x − x0 ) * A + ( y − y0 ) * B + ( z − z0 ) * C = 0 z        p0
If define D = −( Ax0 + By0 + Cz0 ), then
y
Ax + By + Cz + D = 0
x
An Example
Q = [x q        zq ]
T
Find the distance between a point                     yq          and a plane
p = p0 + ur + ws ( 0 ≤ u ≤ 1, 0 ≤ w ≤ 1) . That is to say, find the
ˆ    ˆ

projection of point Q onto plane P and the distance D.
-
Q

D

^
n
-
^
s    ^                    Q’
-      r
po                 -
p
z
y
x
-
Q
Solution
D
^
n
∵ p + QQ ' = Q                            ^
-
s     ^                   Q’
^       ^   ^           -       r
po
∴ p0 + u r + w s + D n = Q
-
p
z
^      ^       ^
y
u r + w s + D n = Q − p0                                        x

⎡u ⎤             ⎡ rx   sx     nx ⎤ ⎡ u ⎤ ⎡ xq ⎤ ⎡ x0 ⎤
⎡^           ⎤                  ⎢                 ⎥         ⎢ ⎥
s n ⎥ ⎢ w ⎥ = Q − P0                 n y ⎥ ⎢ w ⎥ = ⎢ yq ⎥ − ⎢ y0 ⎥
^ ^

⎢r
⎣            ⎦⎢ ⎥               ⎢ ry   sy           ⎢ ⎥              ⎢ ⎥
⎢D⎥              ⎢ rz          s y ⎥ ⎢ D ⎥ ⎢ z q ⎥ ⎢ z0 ⎥
⎣ ⎦              ⎣      sz         ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦
Planar Surface (defined by plane
equation and boundaries)
⎧
⎪ x = x ( u, w )
⎪
⎨ y = y ( u, w )
⎪                   D B            A
⎪ z = z ( u, w ) = − − y ( u, w ) − x ( u, w )
⎩                   c c            c

Ax + By + Cz + D = 0 is satisfied.

The two parametric equation x(u,w) & y(u,w) together with
the boundaries specified by (u=0, w=0,u=1,w=1) determine
the boundary of the projection of a surface p (u , w ) in the x-y
plane.
A Bounded Region of A Plane
⎧ x=u+w                        Chose the expressions of x and y;
⎪                              z is determined by the plane equation.
⎨ y = −u + w
⎪ z = ..........
⎩                                        ⎧x = w
u =0   ⎨              y=x
y                                        ⎩y = w
⎧ x = 1+ w
1
u =1   ⎨              y = x−2
u=0         w=1                 ⎩ y = −1+ w
x              ⎧ x =u
1      2           w=0     ⎨             y = −x
⎩ y = −u
-1 w=0           u=1
⎧ x = u +1
w =1    ⎨             y = −x + 2
-2             The Region             ⎩ y = −u +1
X+Z=2 passes P1(1,0,1) P2(0,0,2) and P3(1,1,1)
p(u2)
p(u)
p(u1)
Bilinear Surface
u1      u              u2

u1 − u      p (u1 ) − p (u )
∵          =
u1 − u 2   p (u1 ) − p (u 2 )

p (u1 )(u1 − u ) − (u1 − u )( p (u1 ) − p (u 2 )) = p (u )(u1 − u 2

(u − u 2 ) p (u1 ) + (u1 − u ) p (u 2 ) = p (u )(u1 − u 2 )

u2 − u                u − u1
∴ p (u   )   =          p (u 1   )+          p (u 2   )
u 2 − u1             u 2 − u1
Bilinear Surface
p(u1,w2)
p(u1,w1)

p(u2,w1)
(u,w)

p(u2,w2)

⎡ u2 − u ⎤ ⎡ w2 − w ⎤                 ⎡ u2 − u ⎤ ⎡ w − w1 ⎤
p (u, w ) = p (u1 , w1 )⎢                       + p (u1 , w2 )⎢         ⎥ ⎢         ⎥
⎣ u2 − u1 ⎥ ⎢ w2 − w1 ⎥
⎦ ⎣         ⎦               ⎣ u2 − u1 ⎦ ⎣ w2 − w1 ⎦
⎡ u − u1 ⎤ ⎡ w2 − w ⎤                 ⎡ u − u1 ⎤ ⎡ w − w1 ⎤
+ p (u2 , w1 )⎢                       + p(u 2 , w2 )⎢          ⎥ ⎢         ⎥
⎣ u2 − u1 ⎥ ⎢ w2 − w1 ⎥
⎦ ⎣         ⎦               ⎣ u 2 − u1 ⎦ ⎣ w2 − w1 ⎦
Ruled (or Lofted) Surfaces
Here we specify two of the four boundary curves, p (u,0) and p (u,1) . These two
curves can be defined by any of the methods that we discussed (cubic spline,
Bezier, B-spline, NURBS, etc.). Points on the surface are obtained by linear
interpolation.

p (u, w ) = p (u,0)(1 − w ) + p (u,1)w

or       p (u, w ) = p (0, w )(1 − u ) + p (1, w )u

If we choose straight lines for the boundaries, the ruled surface becomes a bilinear
surface.
An Example                  p ( u , w ) = p ( u , 0 )(1 − w ) + p ( u ,1) w

p (u,0 ) and p (u,1) are cubic splines with clamped ends
z
or   c1 = p(u,0)           c2 = p(u,1)

p1 = [0 0 0]         p1 = [1 0 0]
T                    T
w=1

p2 = [0 1 0]         p2 = [1 1 1]
T                  T
y        p(u,1)

p1' = [0 1 1]        p1' = [0 1 1]                     w=0           C2
T                     T
p(u,0)
p2 = [0 1 1]         p2 = [0 1 − 1]          C1
'           T        '                  T
x

For each curve (C1 and C2 – cubic splines):

P(u) = (2u3 − 3u2 +1)P + (−2u3 + 3u2 )P + (u3 − 2u2 + u)P' + (u3 − u2 )P'
0                1                 0              1

```
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