CS 284: CAGD
Lecture #2 -- We 8/30, 2000.

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Preparation:

Rockwood: pp 31-41.

Lecture Topics

Homework Discussion

  • How to Build Genus-2 Object

    Some Important Concepts

  • Hodograph: Plot of velocity vector as fct of t
  • Winding Number of a closed curve around a point
  • Turning Number of a closed curve = winding number of hodograph around origin
  • Cn Continuity: first n derivatives are continuous; curve is n-th order differentiable
  • Gn Continuity: first n-order geometric approximations vary smoothly with t (ignoring parametrization)

    How to Draw Smooth Objects

  • Look at results of "Connect the Dots"
  • What is a Spline (physical, mathematical) ?
  • Interpolating spline goes through the dots.
  • Approximating spline is "pulled towards" the dots.

    Definition of Cubic Bezier Curve

  • A Very Simple Spline ...
  • The Defining Control Points
  • The General Behavior
  • Quadratic Case
  • Cubic Case
  • n-th degree Case
    - - How much can we do with a curve of a particular degree ?
    - - See homework !

    Administrative Intermezzo

  • Companion Course CS 294-3 ===> makes a nice complement to CS284 !
  • Class Roster, Accounts, etc.
  • Student Introductions.

    Bernstein Basis Functions

  • Formula

    Properties of Bezier Curves

  • Endpoint Interpolation
  • Tangent Condition
    - - A closer look at the basis functions.
  • Convex Hull
    - - Why does this hold ?
  • Affine Invariance
    - - Could you prove translation invariance ?
  • Variation Diminishing
    - - Counting "wiggles", line intersections.
    - - When do we get inflection points ?
  • Linear Precision
    - - What happens when all control points fall on a straight line ?

    New Homework Assignment:

    Use Rockwood's Interactive Curve Editor (available from the desktop on the PC's in 349 as "Gacd-lab"). Open the applet shown on page 52 of the book, labelled "Higher Degree Bezier Curves" for the following tasks:
    1. Using a heptic Bezier curve {this is degree 7, order 8; using 8 ctrl pts == different ways of saying the same thing},
      model G-1 continuous {continuous tangent directions} loops of as many different turning numbers {the # of times the tangent vector sweep around 360 degrees} as possible -- at least for turning numbers 0, 1, 2.
    2. What order Bezier curve is needed to make a (G1-smooth) loop of turning number 3 ?
    3. Using the minimum number of control points (=minimum order Bezier), make a G1-continuous "figure-8" Bezier curve with C2-point-symmetry {== 2-fold rotational symmetry around a point that will bring the figure back onto itself after a 180-degree rotation around this point}.

    DUE: WED 9/6/00, 9:10am.
    Hand in: window snapshots showing your solutions;
    label your figures with their turning numbers;
    put your name on your hand-ins
    add explanatory comments as necessary.

    On the PCs you can hit Alt+PrnScrn to capture the current active window to the clipboard. You can then paste the clipboard into a program such as "paint" and form there readily send it to the printer.

    Next Reading Assignment:

    Rockwood: pp 42-58.

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