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Study Guide
In addition to the general topics listed below, be able to do all
homework problems, including those ``not to hand in''. In general,
where formulas are developed you should know them, and if there
is a proof in a homework problem you should know it. The exam will
cover the whole course, but will have an extra emphasis on topics
past the midterm.
- Review of vectors and linear transformations
- The projection of a vector on a line
- Cross products
- Relation to matrices; ``mapping the house''; determinants; invertibility
- Orthogonal matrices and transformations
- Meaning and characterization of orthogonal matrices
- Rotation matrices
- Reflections--enough to do homework problems
- ``Three-step'' method for problems
- Affine transformations
- Affine transformations
- Extended matrices
- Preservation properties
- Mapping triangles and parallelograms
- Idea of mapping a standard object of any sort
- ``Hidden explanation''
(see under homogeneous too)
- Applications to windowing, area of triangle, volumes
- Homogeneous coordinates and the real projective plane
- Homogeneous coordinates
- Projective transformations
- Points at infinity
- The real projective plane
- Geometry of the real projective plane
- Making projective transformations
- How much freedom?
- How to make a specific quadrangle go to a specific quadrangle.
- Three-dimensional projective space
- Affine transformations as projective transformations
- The ``hidden'' or ``secret'' explanation using hlt's in
one dimension higher
- Projections from three dimensions to two
- The setup and main classification
- Characteristics of the main types
- How to calculate projections
- Projections on a slanted viewplane
- Kinds of perspective projections--how to recognize
- Subclassifications and how to recognize them, whether given
as a picture or given by an algebraic description of the object,
viewpoint, and viewplane.
- Rotating given latitude/longitude to north pole. Be clear on
which is latitude and which is longitude!
- Convex sets and convex polyhedra
- Convexity
- The convex hull of a set
- Convex combinations
- Convex polyhedra
- Hidden-surface removal for convex polyhedra
- Interpolation for polynomial parametric curves
- Parametric curves in general; polynomial curves
- Polynomial curves
- Lagrange interpolation (concepts; basis functions)
- Properties of polynomials
- Cubic Bézier curves
- Definitions of Bernstein and Bézier
- Properties of the cubic Bernstein polynomials and Bézier
curves. Know those that were used in homework or mentioned in lecture.
- Applications: loops, arcs, arrows, S-curve, Hermite
- Derivative theorem for Bézier curves (of any degree).
- Cubic spline curves
- Bézier curves with zero second derivative at one end
- Gluing two Bézier curves; A-frames
- B-spline curves and their construction
- Interpolation by splines
- Basis functions, how to calculate
- (Skip other possible end conditions and non-uniform spline curves)
- Applications to animation
- Parametric Surfaces
- Parametric curves and surfaces
- Ruled surfaces; bilinear patch (know)
- (Coons patch-don't need to know)
- Tensor bases
- for interpolated spline surfaces
- for Lagrange surfaces
- (not for Bézier surfaces, even though that was on homework)
- How to invent parametric surfaces, as in lab assignment
- Isoparametric curves
- Normal vector at a point
- Ray-tracing
- The setup and its ingredients
- Tracing a ray for a given pixel
- Phong shading
- (not mirror reflection)
- Linear geometry
- Ways of expressing lines in
R
- Relational form and applications
- Relational form
-
- How to tell whether two line segments cross
, and where
-
- Two-point relational formula
- Crossing of lines, line segments
- Points inside/outside triangles, convex polygons
- Points inside non-convex polygons--if discussed
- Labs -- Look them over.
Next: About this document ...
Up: nn_guide
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Kirby A. Baker
2002-03-13