Study guide for midterm

Also see all homework problems (``to hand in'' and ``not to hand in'') and see the text.

• Spaces and maps
  • $\mathbb{R}^n$, $\mathbb{E}^n$

  • parametric lines

  • linear combinations
    • barycentric

  • translations

  • linear maps
    • relation to matrices

    • rotations in 2 dimension

    • reflections

    • meaning of determinant

  • affine maps
    • compatibility with barycentric

  • 3-step problems
    • plane rotations about any center

    • plane reflections using any mirror line

• Curves
  • ways to represent
    • graph of function

    • graph of relation

    • parametric

  • Bézier curves
    • Bernstein polynomials

    • basic properties

    • applications
      • loop

      • arc, arrow

      • S-curve

      • Hermite interpolation

    • de Casteljau algorithm
      • idea of repeated weighted averages

      • geometrically

      • recursion formula

      • table version for computation

    • derivatives
      • at ends; anywhere

    • Bézier of any degree

  • polynomial parametric curves in general
    • basic properties of polynomials

    • Lagrange interpolation
      • theorem

      • proof of uniqueness

      • Lagrange basis functions (blending functions)

      • constructive proof of existence

      • parametric polynomial interpolation

• Surfaces
  • ways to represent
    • graph of function

    • graph of relation

    • parametric

  • examples of surfaces
    • inventing surfaces step-by-step
      • sphere

      • cylinder

      • Möbius strip

      • etc.

    • linear blending of two curves

    • tensor-product surfaces
      • Lagrange

      • Bézier

    • bilinear patch

    • Coons patch

    • normals