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Geometric Modeling

Geomertic Modeling I - Center for Graphics and Geometric Computing, Technion

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An Example

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Outline

 Objective: Develop methods and algorithms to mathematically model shapes of real world objects  Categories:

Wire-frame representations Boundary representations

Primitive based Freeform based Meshes & subdivision

Volumetric representations

Freeform based Voxel Based Octree Based Trivariate Functions

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Wire-Frame Representation

 Object is represented as as a set of points and edges (a graph) containing topological information.  Used for fast display in interactive systems.  Can be ambiguous:

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Ambiguity

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Volumetric Representation

 Voxel based

Space subdivided into equal size boxes – each has on/off flag according to if inside/outside object Scalar values (e.g. density) per voxel are also common Common in imaging applications (CT,MRI,Ultrasound) Memory consuming

 Extension – Octree (recursive construction).  Advantage: simple and robust Boolean operations, in/out tests, can represent and model the interior of the object.  Disadvantage: memory consuming, non-smooth, time consuming to manipulate.

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Display – Iso surfaces

 Difficulty to display volumetric content.  Instead we can represent the boundary of the volume, solving for T(u, v, w) = T0  Marching cubes – examine each individual cube (eight neighboring voxels) and locally decide the shape of the iso surface there.  Result is a polygonal approximation of the iso surface.

• 3
• 3
• 3

+3

• 3

+3

• 3

+3

• 3

+3

• 4

+8 +8 +8

• 8
• 8

T0 = 0

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Constructive Solid Geometry

 Use set of volumetric primitives

Box, sphere, cylinder, cone, etc…

 For constructing complex objects use Boolean operations

Union Intersection Subtraction Complement

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CSG Trees

 Operations performed recursively  Final object stored as sequence (tree) of

• perations on

primitives  Common in CAD packages –

mechanical parts fit well into primitive based framework

 Can be extended with free-form primitives

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Freeform Representation

 Explicit form: z = z(x, y)  Implicit form: f(x, y, z) = 0  Parametric form: S(u, v) = [x(u, v), y(u, v), z(u, v)]  Example – origin-centered sphere of radius R: Explicit is a special case of implicit and parametric form

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A representation of choice

What makes a good representation for solid modeling?

 Ease/Intuitive shape manipulation  Class of representable models  Ease of display and evaluation  Ease of query answering such as inside/outside, subdivision, interference  Robustness and accuracy  Closure under operations such as Booleans.

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Intuitive Curve Editing

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Splines – Free Form Curves

 To allow control of curves need description via set of basis functions + coefficients  Require geometric meaning of coefficients (base)

Approximate/interpolate set of positions, derivatives, etc..

 Typically parametric

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Polynomial Bases

 Monomial basis {1, x, x2, x3, …}

Coefficients are geometrically meaningless Manipulation is not robust

 Number of coefficients = polynomial rank  We seek coefficients with geometrically intuitive meanings  Polynomials are easy to analyze, derivatives remain polynomial, etc.  Other polynomial bases (with better geometric intuition):

Lagrange (Interpolation scheme) Hermite (Interpolation scheme) Bezier (Approximation scheme) B-Spline (Approximation scheme)

 A trajectory of a particle in Rd (planar/3-space/etc.).  Single parameter t [t1,t2] – acts like “time”.  Position = p(t) = (x(t), y(t)), or p(t) = (x(t), y(t), z(t)), Tangent velocity = v(t) = (x’(t), y’(t) {,z’(t)})  Examples:

p(t) = (cos(t), sin(t)) t  [0,2) ||v(t)|| 1 p(t) = (cos(2t), sin(2t)) t  [0,) ||v(t)|| 2 p(t) = ((1-t2)/(1+t2), 2t/(1+t2)) t  (-,+) ||v(t)|| ?

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Parametric Curves

v(t) = (x’(t),y’(t))

(x(t),y(t))

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Mathematical Continuity

 C1(t) & C2(t), t  [0,1] - parametric curves  Level of continuity of the curves at C1(1) and C2(0) is:

C-1: C1(1)  C2(0) (discontinuous) C0: C1(1) = C2(0) (positional continuity) Ck, k > 0 : continuous up to kth derivative

 Continuity of single curve inside its parameter domain is similarly defined - for polynomial bases it is C

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Geometric Continuity

 Mathematical continuity is too strong a requirement  May be relaxed to geometric continuity

Gk, k  0 : Same as Ck Gk, k =1 : C'1(1) = C'2(0) Gk, k 0 : There is a reparameterization of C1(t) & C2(t), where the two are Ck

 E.g.

C1(t)=[cos(t),sin(t)], t[/2,0] C2(t)=[cos(t),sin(t)], t[0,/2] C3(t)=[cos(2t),sin(2t)], t[0,/4] C1(t) & C2(t) are C1 (& G1) continuous C1(t) & C3(t) are G1 continuous (not C1)

C1 C2, C3

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More on Continuity

 Consider a set of points {Pi}i=0,n  A piecewise constant interpolant will be C-1  A piecewise linear interpolant will be C0  Polyhedra are usually piecewise linear approximation

• f freeform polynomials

 Polyhedra are C0 continuous  Higher order polynomials posses better continuity

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Cubic Hermite Basis

The set of polynomials of degree k is a linear vector space of degree k+1 The canonical, monomial basis for polynomials is {1, x, x2, x3,…} Define geometrically-oriented basis for cubic polynomials hi,j(t): i, j = 0,1, t[0,1] Has to satisfy:

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Hermite Cubic Basis (cont’d)

 The four cubics which satisfy these conditions are  Obtained by solving four linear equations in four unknowns for each basis function  Prove: Hermite cubic polynomials are linearly independent and form a basis for cubic polynomials

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Hermite Cubic Basis (cont’d)

 Lets solve for h00(t) as an example.  h00(t) = a t3 + b t2 + c t + d must satisfy the following four constraints:  Four equations and four unknown to solve for.

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Hermite Cubic Basis (cont’d)

 Let C(t) be a cubic polynomial defined as the linear combination:  What are C(0), C(1), C’(0), C’(1) ?  C(0) = P0, C(1) = P1, C’(0) = T0, C’(1) = T1  To generate a curve through P0 & P1 with slopes T0 & T1 use

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Parametric Splines

 So far we solve for a single explicit function C(t)  Cannot represent closed curves such as a circle  Solution: let Pi = (xi, yi) and Ti = (ti

x, ti y) be vector

functions  Solve for each individual axis independently Here is an example of four Hermite cubics that approximate a circle

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Cubic Splines

 Standard spline input – set of points {Pi}i=0, n

No derivatives’ specified as input Can we prescribe derivatives values in order to achieve C2?

 Interpolate by n cubic segments (4n DOF):

Derive {Ti}i=0 , n from C2 continuity constraints Solve 4n equations

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Cubic Splines

 Have two degrees of freedom left (to reach 4n DOF)  Options

Natural end conditions: C1''(0) = 0, Cn''(1) = 0 Complete end conditions: C1'(0) = 0, Cn'(1) = 0 Prescribed end conditions (derivatives available at the ends): C1'(0) = T0, Cn'(1) = Tn Quadratic end conditions C1''(0) = C1''(1), Cn''(0) = Cn''(1), Periodic end conditions C1'(0) = Cn'(1), C1''(0) = Cn''(1),

 Question: What parts of C(t) are affected as a result of a change in Pi ?

prescribed natural

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 The assumption t  [0,1] (uniform parameterization) is arbitrary

Implicitly implies same curve length for each segment

 Not natural if points are not equally spaced  One alternative - chord-length parameterization:  Question: Can chord-length parameterization be used for cubic Hermite splines?

Parameterization

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Parameterization

Question: What do we need arc/chord length curves for? Uniform Chord- length [0,1] [0,1] [0,8] [0,1] [0,3] [0,1] [0,1] [0,4]

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Bezier Curves

The Bezier curve is an approximation

• f a given control polygon (a polyline)

Denote by  (t): t[0,1] Bezier curve of degree n is defined over n+1 control points, {Pi}i=0, n Have two formulations

Constructive Algebraic

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De Casteljau Construction

 (1/3) Select t[0,1] value. Then,

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Algebraic Form of Bezier Curves

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Algebraic Form of Bezier Curves

Bezier curve for set of control points {Pi}i=0, n : where {Bi

n(t)}i=0, n Bernstein basis of polynomial

• f degree n

Here is a Cubic case example:

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Algebraic Form of Bezier Curves



why?

 Curve is linear combination of basis functions  Curve is affine combination of control points

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Properties of Bezier Curves

  (t) is polynomial of degree n   (t)  CH(P0,…,Pn)   (0) = P0 and  (1) = Pn   '(0) = n(P1P0) and  '(1) = n(PnPn-1)   (t) is affine invariant and variation diminishing   (t) is intuitive to control via Pi and it follows the general shape of the control polygon   '(t) is a Bezier curve of one degree less  Questions:

What is the shape of a Bezier curve whose control points lie on a line? How can one connect two Bezier curves with c0 continuity? c1 ? c2 ?

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Bezier Curve Editing

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Some Trivia

 Bezier curves introduced nearly simultaneously by Pierre Bezier at Renault and Paul De Casteljau at Citroen (late 60s)  Attributed to Bezier as he was the first to make the results public

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 Degree corresponds to number of control points

Global support: change in one control point affects the entire curve For large sets of points – curve deviates far from the points

 Cannot represent conics exactly. Most noticeably circles

Can be resolved by introducing a more powerful representation

• f rational curves.

As an example, a 90 degrees arc as a rational Bezier curve:

P0

=(0,1)

P1

=(1,1)

P2

=(1,0)

Drawbacks of Bezier Curves

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B-Spline Curves

 Idea: Generate basis where functions are continuous across the domains while presenting local support  For each parameter value only a finite set of basis functions is non zero.  The parametric domain is subdivided into sections at parameter values denoted knots, {i }.  The B-spline functions are then defined over the knots.  The knots are denoted uniform knots if i - i-1 = c,

• constant. W.l.o.g.,

assume c = 1.

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Uniform Cubic B-Spline Curves

 Definition (uniform knot sequence, i - i-1 = 1):

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Uniform Cubic B-Spline Curves

 For any t  [3, n]: (prove it!)  For any t  [3, n] at most four basis functions are non zero  Any point on a cubic B-Spline is an affine combination

• f at most four control points

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 In general, B-Splines do not interpolate control points

in particular, the uniform cubic B-spline curves do not interpolate the end points of the curve. Why is the end points’ interpolation important?

 Two ways are common to force endpoint interpolation:

Let P0 = P1 = P2 (same for other end) Add a new control point (same for other end) P-1 = 2P0 – P1 and a new basis function N-1

3(t).

 Question:

What is the shape of the curve at the end points if the first method is used? What is the derivative vector of the curve at the end points if the first method is used?

Boundary Conditions for Cubic B-Spline Curves

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Local Control of B-spline Curves

Control point Pi affects  (t) only for t(i, i+4)

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Intuitive B-spline Curve Editing

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Properties of B-Spline Curves

  For n control points,  (t) is a piecewise polynomial of degree 3, defined over t[3, n)  and for specific t = t0?  (t) is affine invariant and variation diminishing  (t) follows the general shape of the control polygon and it is intuitive and ease to control its shape  Questions:

What is  (i) equal to? What is  '(i) equal to? What is the continuity of  (t) ? Prove it!

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From Curves to Surfaces

 A curve is expressed as inner product of coefficients Pi and basis functions  One way of extending curves into surfaces is to treat surface as a curves of curves. This approach is also known as tensor product surfaces  Assume Pi is not constant, but are functions of a second, new parameter v:

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From Curves to Surfaces (cont’d)

Then

• r

Question: What is the difference between u and v?

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Intuitive Surface Editing

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Basic Surface Properties

Partials (Tangents) and Normal Elliptic regions (Convex/Concave)

S(u, v) S(u0, v) S(u, v0) Isoparametric Curves

Hyperbolic regions (Saddle-like)

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Surface constructors

 Construction the geometry is a first stage in any synthetic image synthesis process  There exist a set of high level, simple and intuitive, surface constructors for novice users:

Bilinear patch Ruled surface Boolean sum Surface of Revolution Extrusion surface Surface from curves (skinning) Swept surface

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Bilinear Patches

 Bilinear interpolation of 4 3D points - 2D analog of 1D linear interpolation between 2 points in the plane  Given P00, P01, P10, P11 the parametric Bilinear surface for u,v[0,1] is:  Questions:

What does an isoparametric curve of a Bilinear patch look like ? Can you represent the Bilinear patch as a Bezier surface? When is a Bilinear patch planar?

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Ruled Surfaces

 Given two curves a(t) and b(t), the corresponding ruled surface between them is constructed by

The corresponding points on a(t) and b(t) are connected by straight lines

 Questions:

When is a ruled surface a bilinear patch? When is a bilinear patch a ruled surface?

S(u, v) = v a(u) + (1-v) b(u)

a(u) b(u)

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Ruled Surfaces

http://www.bvd.co.il

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Boolean Sum

 Given four connected curves i i=1,2,3,4, Boolean sum S(u, v) fills the interior. Let, Then

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Boolean Sum (cont’d)

 S(u,v) interpolates the four  i along its boundaries.  For example, consider the u = 0 boundary: Question: Must i be coplanar?

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Surface of Revolution

 Rotate a, usually planar, curve around an axis  Consider curve (t) = (x(t), 0, z(t)) and let Z be the axis of

• revolution. Then,

Question: sin and cos are not rational. Can we represent surfaces of revolution as rational?

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Extrusion Surface

 Extrusion of a, usually planar, curve along a linear segment.  Consider curve (t) and vector  Then

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Surface from Curves (Skinning)

 An intuitive surface fitting (approximation or interpolation) scheme  Fit a surface through a set of prescribed curves: Question: Must the curves be planar? Closed?

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Sweep Surface

 A general surface construction scheme  Rigid motion of one (cross section) curve along another (axis) curve:  The cross section can change as it is swept along Question: Is an Extrusion as special case of sweep? Surface of revolution?

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Quiz

What surface constructors were used to create these models: