disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004

SURFACE

disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004

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

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Taxonomy of surfaces for CAD and CG

2. Simple basic surface

- Sphere, Cube, Cone, and Cylinder

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Taxonomy of surfaces for CAD and CG

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.

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Taxonomy of surfaces for CAD and CG

3. Ruled surface (cont)

C1C2 C1

C2

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Taxonomy of surfaces for CAD and CG

3. Tabulated cylinder

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

Shapecurve

Vector

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Taxonomy of surfaces for CAD and CG

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

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Taxonomy of surfaces for CAD and CG

4. Surface of revolution (cont)

curve

axis

disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004

Taxonomy of surfaces for CAD and CG

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)

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Taxonomy of surfaces for CAD and CG

5. Swept surface (cont)

Shape curve

Path- an arbitrary curve

disediakan oleh Suriati bte Sadimon GMM, FSKSM, UTM 2004

Taxonomy of surfaces for CAD and CG

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

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Taxonomy of surfaces for CAD and CG

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

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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

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NURBS Surface

P0,0

P0,1

P0,2

P0,3

P1,0 P2,0 P3,0

P3,3

u

v

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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

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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

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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

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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

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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

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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)

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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.• .

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Example- curve

•subdivision for curve(bezier) in the plane

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Example - surface

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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