(Smooth) CURVES and SURFACES

Lecture #2  for CS 274

Wednesday  3/16, 2005 -- 12:40-2:00pm, -- 203 McLaughlin

Differential Geometry of Curves

REFERENCE -- TO LEARN MORE:

"Analysis" of Curves (as compared to "Design" or "Data Fitting")

• Intrinsic Curve Properties -- defined at each local point on the curve
- - Vectors: Tangent, Normal, Binormal;
- - Planes: Osculating, Normal, Rectifying;
- - Scalars: Curvature, Torsion.
• Extrinsic Curve Properties -- overall, global values
- - Arc-length;
- - Planarity, Linearity;
- - Closedness, Turning Number, Knot Type;
- - Highest polynomial degree, lowest continuity type, # of inflection points.

Construction of the Frenet Frame (Ref. frame for intrinsic properties)

• Finding the Tangent: -- p' -- velocity vector -- normalize ...
• Acceleration: -- p" -- change of velocity vector: 2 components (true acceleration, turning) ...
• Normal Plane: -- normal to tangent vector.
• Osculating Plane: -- containing osculating circle = the one that best (locally) fits the curve.
• Normal vector: -- indicates direction in which curve bends.
• Binormal: -- axis around which Frenet frame turns.
• Difference between Normal and second derivative ...
• The three coordinate planes and their relations to the 3 vectors ...

Serret - Frenet Relations

• What is Curvature ? -- rate of rotation around b (1/r of the osculating circle)
• What is Torsion ? -- rate of rotation of osculating plane (around tangent)
• Inflection points -- where curvature = 0.
• Which unit vectors serve as local rotation axis ?  (==> see above)
• Which unit vectors could best serve as graphical indicators (along curve)?
-- for curvature: use -n; {n would crowd the inside of the bend} ==> "hedgehog" curves.
-- for torsion: use b; { allows both signs; t would crowd the tangent vector}

Puzzle

• Given: curvature as a function of arc-length, -- does this define the shape of a curve ?

Derivatives of Splines

• Derivatives of polynomial expressions -- polynomials of one degree less
  
• Differentiating Bernstein polynomials -- leads to other Bernstein polynomials ==> Nice recurrence relations ...
  
• A geometrical view: The Hodograph of the Bezier curve
• Finding the Control Points for the differentiated Bezier curve
• Derivatives for B-Splines
• Recurrence relations for uniform B-Splines

REFERENCES -- TO LEARN MORE:

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Introduction to Subdivision

Interpolating mid-segment subdivisions
 -- Find new mid-points for each linear segment
     -- properly bulging out to lie on curve implied by nearest neighbors;
     -- new points, once placed, stay in same location forever.

• Linear interpolation  (is this useful ?)
• Add a random disturbance  (what is this good for?)
• To make a smooth curve, we should try a quadratic interpolant  (what is the difficulty?)
• Do one on the left and one on the right and average  (what is the result?)
• Try a cubic interpolant  (what are the needed weights?)
• Circular interpolants  (what guarantees can we make?)

An Approximating Curve Subdivision Scheme

Surface Subdivision Schemes (Two-Manifolds)

REFERENCES -- TO LEARN MORE: