Curves and Surfaces

Curves and Surfaces

Mathematical curves and surfaces allow smooth, continuous shapes to be defined by control points. They are essential for modeling organic shapes, fonts, and smooth surfaces.

Bezier Curves

Defined by control points. The curve starts at the first control point and ends at the last, with intermediate points pulling the curve toward them.

Quadratic Bezier (3 control points):

  B(t) = (1-t)²P0 + 2(1-t)tP1 + t²P2    t ∈ [0,1]

  P0 ---- P1 ---- P2
    \           /
     \         /     Curve pulled toward P1
      \       /
       \     /
        -------

Cubic Bezier (4 control points):

  B(t) = (1-t)³P0 + 3(1-t)²tP1 + 3(1-t)t²P2 + t³P3

  P0 ---- P1
    \     /
     \   /         More flexible shape control
      \ /
       X
      / \
     /   \
  P3 ---- P2

De Casteljau's Algorithm

A numerically stable method for evaluating Bezier curves through repeated linear interpolation.

Cubic Bezier evaluation:

  Given P0, P1, P2, P3 and parameter t:

  Level 1:
    Q0 = lerp(P0, P1, t)
    Q1 = lerp(P1, P2, t)
    Q2 = lerp(P2, P3, t)

  Level 2:
    R0 = lerp(Q0, Q1, t)
    R1 = lerp(Q1, Q2, t)

  Level 3:
    S0 = lerp(R0, R1, t)   ← point on curve at t

  lerp(a, b, t) = (1-t)*a + t*b

B-Splines

Basis splines offer more control than Bezier curves. They are piecewise polynomial curves that pass through control points with continuity constraints.

B-Spline properties:
  - Local control: moving one control point affects
    only a portion of the curve
  - Higher continuity at knots
  - Defined by control points + knot vector

Cubic B-Spline basis functions:
  N0(t) = (1-t)³/6
  N1(t) = (3t³ - 6t² + 4)/6
  N2(t) = (-3t³ + 3t² + 3t + 1)/6
  N3(t) = t³/6

  S(t) = Σ Ni(t) * Pi

NURBS

Non-Uniform Rational B-Splines extend B-splines with weights, allowing exact representation of conic sections (circles, ellipses, hyperbolas).

NURBS curve:

  C(t) = Σ Wi * Ni(t) * Pi
         -------------------
           Σ Wi * Ni(t)

  Wi = weight of control point Pi
  Ni = B-spline basis function
  Pi = control point

  Weight > 1 → curve pulled toward Pi
  Weight = 1 → standard B-spline
  Weight < 1 → curve pushed away from Pi

Used in: CAD, 3D modeling, animation (Maya, 3ds Max)

Surfaces

Bézier Surface (tensor product):

  S(u,v) = Σ Σ Bi,m(u) * Bj,n(v) * Pij

  Where Bi,m are Bezier basis functions
  Pij is the control point grid (m+1 x n+1)

NURBS Surface:

  S(u,v) = Σ Σ Wi,j * Ni(u) * Nj(v) * Pij
           -------------------------------
             Σ Σ Wi,j * Ni(u) * Nj(v)