interpolation and splines

• interpolating curves pass exactly through specified data points (knots)
• approximating curves do not necessarily pass through knots, but minimize some error or energy functional
• hybrid approaches exist, where only selected knots are interpolated

continuity is typically expressed as ck continuity.

• c0: positional continuity only
• c1: continuous first derivative (tangent continuity)
• c2: continuous second derivative (curvature continuity)
• higher orders are possible but rarely justified outside numerical analysis higher continuity generally implies:
• wider support (each segment depends on more knots)
• reduced local control
• increased sensitivity to knot placement

• local schemes change only nearby segments when a knot moves
• global schemes change the entire curve when a single knot moves global control simplifies optimization but complicates interactive design.

many interpolants overshoot.

• values between knots may exceed the convex hull or bounding box of the knots
• this is undesirable for monotonic data, physical constraints, or safety-critical systems
• avoiding overshoot often conflicts with high-order continuity bounding properties are therefore a primary selection criterion.

the curve parameter is rarely unique.

• uniform parameterization
• chord-length parameterization
• centripetal parameterization parameterization affects speed, curvature, and overshoot, even for the same basis functions.

piecewise low-degree polynomials with continuity constraints at knots.

• cubic hermite splines

  • c1 or c2 continuity depending on tangent choice
  • explicit tangent control

• catmull-rom splines

  • interpolating
  • typically c1
  • local control
  • prone to overshoot unless modified

• natural cubic splines

  • interpolating

  • c2 continuity

  • global solution

  • second derivative constrained to zero at endpoints

polynomial curves defined by control points rather than interpolation points.

• bezier curves

  • do not interpolate intermediate control points
  • strong convex hull and bounding box properties
  • degree grows with number of control points

• composite bezier splines

  • piecewise bezier with continuity constraints

  • manual continuity management

splines defined by basis functions with compact support.

• b-splines

  • non-interpolating (except at ends, depending on knot vector)
  • ck continuity controllable via knot multiplicity
  • strong local control

• nurbs

  • rational generalization of b-splines

  • exact representation of conic sections

  • weights introduce additional degrees of freedom

  • same locality and continuity structure as b-splines

curves constructed from geometric primitives rather than polynomials.

• circular arc interpolation

  • exact curvature control
  • c1 continuity at joins unless constrained
  • common in cnc and motion planning

• spherical linear interpolation (slerp)

  • interpolation on manifolds

  • preserves constant angular velocity

  • not polynomial

interpolation performed in transformed spaces.

• exponential interpolation

  • useful for scale-like quantities
  • avoids negative values by construction
• log-space or power-law interpolation: preserves ratios rather than differences these methods trade linear intuition for constraint preservation.

curves defined as minimizers of a functional.

• thin-plate splines
• smoohing splines
• minimum curvature or minimum acceleration curves properties:
• typically global
• interpolating or approximating depending on constraints
• physically interpretable but computationally heavier