[Computer Animation] Bezier Curve & Bezier Spline

Bezier Curve & Bezier Spline

Basic Functions

• A linear space of cubic polynomials
  • Monomial basis $(t^{3}, t^{2}, t^{1}, t^{0}$
    • $x(t) = a_{3}t^{3} + a_{2}t^{2} + a_{1}t + a_{0}$
  • The coefficients $a_{i}$ do not give tangible(유형의) geometric meaning.

 

Bezier Control Points

• Control Points 
  • $b_{0}, b_{1}, b_{2}, b_{3}$
• Demo
  • http://blogs.sitepointstatic.com/examples/tech/canvas-curves/bezier-curve.html

Canvas Bézier Curve Example

 

Bezier Curve(베지어 곡선)

• Cubic polynomial in Bernstein Bases(번스타인 기초)
• $p(t) = b_{0}B_{0}^{3}(t) + b_{1}B_{1}^{3}(t) + b_{2}B_{2}^{3}(t) + b_{3}B_{3}^{3}(t)$ $(0 ≤ t ≤ 1)$
• Bernstein Basis Functions

 

Cubic Bernstein Basis Functions

 

Bernstein Basis Functions

 

 

• $p(t)$ is a linear combination of points $b_{0}, b_{1}, b_{2},$ and $b_{3}$

 

• Derivative(미분) of $p(t)$

 

Properties of Cubic Beizer Curves

• End point interpolation
  • $p(0) = b_{0}$
  • $p(1) = b_{3}$
• The tangent vectors to the curve at the end points are coincident with the first and last edges of the control point polygon.
  • $p'(0) = 3(b_{1} - b_{0})$
  • $p'(1) = 3(b_{3} - b_{2})$

 

• Invariance under Affine Transformation
  • Partition of unity of Bernstein Basis functions
• The curve is contained in the Conven Hull of the control polygen
• Variation Diminishing
  • The curve in 2D space does not oscillate about any straight line more often than the control point polygon.

 

Bezier Spline

Beizer Splines

• When four key points and tangent conditions are given,

 

What is the First Bezier Curve?

 

First Bezier Curve?

 

Bezier Spline