Curves and Surfaces Modeling

• Curves and surfaces are discussed.
• Modeling and approximations of curves and surfaces are introduced.

Escaping Flatland

• Previously worked with flat entities, like lines and flat polygons.
• Flat entities fit well with graphics hardware and are mathematically simple.
• The world is not composed of flat entities.
• Curves and curved surfaces are needed.
• Curves and curved surfaces may only have a need at the application level.
• Implementation can render curves and surfaces approximately with flat primitives.

Modeling with Curves

• Curves can be represented by interpolating data points.
• Curves can be represented by approximating curves.

What Makes a Good Representation

• There are many ways to represent curves and surfaces.
• A good representation should be stable and smooth.
• A good representation should be easy to evaluate.
• One must consider whether to interpolate or come close to data.
• One must consider if derivatives are required.

Explicit Representation

• The dependent variable is given explicitly in terms of the independent variable.
• In 2D, the most familiar form of a curve is y=f(x).
• Explicit representations cannot represent all curves, such as vertical lines and circles.
• An extension to 3D includes y=f(x), z=g(x).
• The form z = f(x,y) defines a surface
• The graph of a function is usually described by an equation and is called an explicit representation.

More on Explicit Representation

• Curves in 2D can be represented as y=f(x).
• Curves in 3D can be represented as y=f(x), z=g(x).
• Surfaces in 3D can be represented as z=f(x,y).
• Vertical lines where x = c cannot be represented as y = f(x).
• Circles can be represented as y = ±(r² – x2)1/2, giving two or zero values for x.
• Explicit representations are too dependent on coordinate systems.
• Explicit representations are rarely used in computer graphics.

Implicit Representation

• The dependent variable has not been given explicitly in terms of the independent variable.
• Two dimensional curves can be represented as g(x,y)=0.
• Implicit curves are much more robust.
• All lines can be represented as ax+by+c=0.
• Circles can be represented as x²+y²-r²=0.
• Three dimensions are g(x,y,z)=0, which defines a surface.
• Two surfaces intersect to get a curve.
• Generally, it is not possible to solve for points that satisfy implicit equations.
• An implicit surface is the set of zeros of a function of three variables where the equation is not solved for x, y, or z.

Algebraic Surface

• An algebraic surface is an algebraic variety of dimension two.
• In geometry over complex numbers, an algebraic surface has complex dimension two (as a complex manifold when non-singular).
• A quadric surface satisfies 2 ≥ i+j+k.
• You can represent algebraic surfaces using vectors, scalars, and a type identifier.
• They have at most 10 terms.
• One can solve for the intersection with a ray by reducing the problem to solving a quadratic equation.
• Both explicit and implicit curves are nonparametric.

More on Implicit Representation

• Curves in 2D can be represented as f(x,y)=0.
• Lines can be represented as ax+by+c=0.
• Circles can be represented as x² +y² -r² =0.
• Surfaces in 3D can be represented as f(x,y,z)=0.
• Planes can be represented as ax+by+cz+d=0.
• Spheres can be represented as x² +y² + +z² -r² =0.
• f(x,y,z) can describe a 3D object based on the following conditions:
• Inside: f(x,y,z) < 0
• Surface: f(x,y,z) = 0
• Outside: f(x,y,z) > 0

Parametric Curves

• Parametric curves use a separate equation for each spatial variable.
• x=x(u), y=y(u), and z=z(u), where p(u)=[x(u), y(u), z(u)]T.
• Each value of (u) defines a point p(u) that is plotted. The collection of all points results in a parametric curve.
• For umax  u  umin you can trace out a curve in two or three dimensions.

Selecting Functions

• One can select good functions.
• These functions would not uniquely define a spatial curve.
• Can approximate or interpolate known data.
• Functions should be easy to evaluate and differentiate.
• Functions should allow computation of normals and connecting pieces (segments) easily.
• Functions should be smooth.

Parametric Lines

• The parameter u can be normalized over the interval (0,1).
• A line connecting two points p0 and p1 is defined as p(u)=(1-u)p0+up1.
• A ray from p0 in the direction d is defined as p(u)=p0+ud.

Parametric Surfaces

• Surfaces require two parameters.
• x=x(u,v), y=y(u,v), z=z(u,v) results in p(u,v) = [x(u,v), y(u,v), z(u,v)]T.
• parametric surfaces have the same properties as curves.
• Smoothness
• Differentiability
• Ease of evaluation

Normals

• You can differentiate with respect to u and v to obtain the normal at any point p.

Parametric Planes

• Point-vector form: p(u,v)=p0+uq+vr where n=qxr.
• Three-point form: q = p1 – p0 and r = p 2 – p0.

Parametric Sphere

• x(u,v) = r cos  sin 
• y(u,v) = r sin  sin 
• z(u,v) = r cos  cos 
• 360    0 and 180    0
•  constant: circles of constant longitude
•  constant: circles of constant latitude
• Differentiate to show n = p.

Curve Segments

• After normalizing u, each curve is written as p(u)=[x(u), y(u), z(u)]T, 1  u  0.
• In classical numerical methods, we design a single global curve.
• In computer graphics and CAD, it is better to design small connected curve segments.

Parametric Polynomial Curves

• If N=M=L, we need to determine 3(N+1) coefficients.
• This is equivalent to 3(N+1) independent conditions (Ck).
• Given curves for x, y, and z are independent, we can define each independently in an identical manner. n
• Use the form where p(u)=[x(u), y(u), z(u)] where p can be any of x, y, or z.

Why Polynomials

• Easy to evaluate.
• Continuous and differentiable everywhere.
• Must worry about continuity at join points, including continuity of derivatives.

Cubic Parametric Polynomials

• N=M=L=3, gives balance between ease of evaluation and flexibility in design.
• Four coefficients to determine for each of x, y, and z.
• Seek four independent conditions for various values of u resulting in 4 equations in 4 unknowns for each of x, y, and z.
• Conditions are a mixture of continuity requirements at the join points and conditions for fitting the data.

Cubic Polynomial Surfaces

• p(u,v)=[x(u,v), y(u,v), z(u,v)]T where p is any of x, y, or z.
• Need 48 coefficients (3 independent sets of 16) to determine a surface patch.

Surface Patch

• A surface may be composed of one or more patches.
• Each patch has its own U-V coordinate system.