CHAPTER 12 - Curves and Surfaces
Objectives
- Introduce types of curves and surfaces
- Explicit
- Implicit
- Parametric
- Strengths and weaknesses
- Discuss Modeling and Approximations
- Conditions
- Stability
Escaping Flatland
- Until now we have worked with flat entities
such as lines and flat polygons
- Fit well with graphics hardware
- Mathematically simple
- But the world is not composed of flat entities
- Need curves and curved surfaces
- May only have need at the application level
- Implementation can render them approximately with flat primitives
Modeling with Curves
What Makes a Good Representation?
- There are many ways to represent curves and surfaces
- Want a representation that is
- Stable
- Smooth
- Easy to evaluate
- Must we interpolate or can we just come close to data?
- Do we need derivatives?
Explicit Representation
- Most familiar form of curve in 2D
- Cannot represent all curves
- Vertical lines
- Circles
- Extension to 3D
- The form z = f(x,y) defines a surface
y=f(x)
y=f(x), z=g(x)
Implicit Representation
- Two dimensional curve(s)
- Much more robust
- All lines ax+by+c=0
- Circles x2+y2-r2=0
- Three dimensions g(x,y,z)=0 defines a surface
- Intersect two surfaces to get a curve
- In general, we cannot solve for points that satisfy
g(x,y)=0
Algebraic Surface
- Quadric surface 2 >= i+j+k
- At most 10 terms
- Can solve intersection with a ray by reducing problem to solving quadratic equation
Parametric Curves
Selecting Functions
- Usually we can select “good” functions
- Not unique for a given spatial curve
- Approximate or interpolate known data
- Want functions which are easy to evaluate
- Want functions which are easy to
differentiate
- Computation of normals
- Connecting pieces (segments)
- Want functions which are smooth
Parametric Lines
Parametric Surfaces
Normals
Parametric Planes
Parametric Sphere
Curve Segments
Parametric Polynomial Curves
Why Polynomials?
Cubic Parametric Polynomials
Cubic Polynomial Surfaces
