disediakan oleh suriati bte sadimon gmm, fsksm, utm 2004 surface

disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004

SURFACE


Taxonomy of surfaces for CAD and CG

1. Plane surface

- the most elementary of the surface type

- defined by four curves/ lines or by three points or a line and a point



2. Simple basic surface

- Sphere, Cube, Cone, and Cylinder



3. Ruled surface

• produced by linear interpolation between two bounding geometric elements. (curves, c1 and c2)

• Bounding curves must both be either geometrically open (line, arc) or closed (circle, ellipse).

• a surface is generated by moving a straight line with its end points resting on the curves.



3. Ruled surface (cont)

C1C2 C1

C2



3. Tabulated cylinder

• Defined by projecting a shape curve along a line or a vector

Shapecurve

Vector



4. Surface of revolution

• Generated when a curve is rotated about an axis

• Requires –• a shape curve (must be continuous)• a specified angle• an axis defined in 3D modelspace.

• The angle of rotation can be controlled

• Useful when modelling turned parts or parts which possess axial symmetry



4. Surface of revolution (cont)

curve

axis



5. Swept surface

• A shape curve is swept along a path defined by an arbitrary curve.

• Extension of the surface of revolution (path a single curve) and tabulated surface (path a vector)



5. Swept surface (cont)

Shape curve

Path- an arbitrary curve



6. Sculptured surface • Sometimes referred to as a “curve mesh” surface.• coon’s patch• among the most general of the surface types• unrestricted geometric• Generated by interpolation across a set of defining shape curves



6. Sculptured surface (cont)

• Or

• A set of cross-sections curves are established. The system will interpolate the crosssections to define a smooth surface geometry.

• This technique called lofting or blending


NURBS Surface

– P(u,v) = wi,jNi,k(u) jNj,l(v) pi,j

wi,jNi,k(u) jNj,l(v)– u, v = knot values in u and v direction (u k-1 u un+1 ,v

k-1 v vn+1)– pi,j - control points (2D graph)– Degree = k-1 (u direction) and l–1 (v direction)– wi,j – weights (homogenous coordinates of the control

points)

i=0 j=0

n m

i=0

n

j=0

m


NURBS Surface

P0,0

P0,1

P0,2

P0,3

P1,0 P2,0 P3,0

P3,3

u

v


Normal Vector

• Perpendicular to the surface

• aTangent vector in u direction.

• b tangent vector in v direction.

• Normal vector, n = a x b (cross product)

• a = dP(u,v) b = dP(u, v)

du dv

Na

b


NURBS surface generated by sweeping a curve

• Example – sweep along a vector/ line

• NURB curve, P has degree l-1, knot value (0,1,…m) and control points Pj

• Sweep along a line translate the curve in u direction.• direction linear degree = 1 2 control points

knot value = 0,0,1,1

Pj

uv

d

a


NURBS surface generated by sweeping a curve

• Example – sweep along a vector/ line

• P0, j = P j , P1, j = P j + da, h0, j = h1, j = h j

• NURBS equation

– P(u,v) = wi,jNi,2(u) jNj,l(v) pi,j

wi,jNi,2(u) jNj,l(v)

Pj

uv

d

a

1

i=0

m

j=0


NURBS surface generated by revolving a curve

x

z

v

y

xP0, j = P 8, j = Pj

P1, jP2, jP3, j

P4, j

P5, j P6, j P7, j

Pj • v direction NURBS curve, P has degree = l-1, control points Pj, •Revolution axis = z axis• u direction circle 9 control points degree = 2 knot vector (0,0,0,1,1,2,2,3,3,4,4,4)

•P0, j = P j , h0, j = h j

•P1, j = P0,j+ x j j, h1, j = h j .1/2•P2, j = P1,j- x ji, h2, j = h j

•P3, j = P2,j- x j i, h3, j = h j .1/2

u


NURBS surface generated by revolving a curve

x

z

v

y

xP0, j = P 8, j = Pj

P1, jP2, jP3, j

P4, j

P5, j P6, j P7, j

Pj •P4, j = P3,j- x j j, h4, j = h j

•P5, j = P4,j- x jj, h5, j = h j 1/2•P6, j = P5,j- x j i, h6, j = h j

•P7, j = P6,j- x ji, h7, j = h j 1/2•P8, j = P0,j, h8, j = h j

•NURBS equation•P(u,v) = wi,jNi,3(u) jNj,l(v) pi,j

wi,jNi,3(u) jNj,l(v)

u

8

i=0

m

i=0


NURBS surface display• Use simple basic surface

– Mesh polygon – flat faces triangle / rectangle

• Patches– A patch is a curve-bounded collection of points

whose coordinates are given by continuous, two parameter, single-valued mathematical functions of the form

– x = x(u,v) y= y(u,v) z = z(u,v)


Idea of subdivision• Subdivision defines a smooth curve or surface as the limit of a sequence of

successive refinements.• The geometric domain is piecewise linear objects, usually polygons or polyhedra.• .


Example- curve

•subdivision for curve(bezier) in the plane


Example - surface


Benefit of subdivision

• The benefit – simplicity and power

• Simple – only polyhedral modeling needed, can be produced to any desired tolerance, topology correct

• Power – produce a hierarchy of polyhedra that approximate the final limit object

disediakan oleh suriati bte sadimon gmm, fsksm, utm 2004 surface

Documents

surface slide

taxonomy of surfaces

curve mesh surface

plane surface

u j n j

arbitrary curve slide

tabulated surface path

sculptured surface cont