Mathematics 4270
Fall Semester, 1999

Topic list for the exam on October 20. This is a closed-book
exam. It takes place during the usual class period, in the
usual room.

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2.1 all. Esp the program to draw a curve

2.2  DVI language. Know how to use the basic features. These
include 2.5.

2.3  A general understanding is a good idea, but you will
not be tested on the detailed features of this program.

2.4  Know how to interpret logo code. 

2.5 See above.

2.6 Know how to interpret programs like those on pp 115-6.

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3.1.2 Derive the equation of an ellipse. The derivation
(p. 140) Parametric
equations for an ellipse, in Lemmas 5 and 6.

3.1.3 Derive the equation of a hyperbola. Parametric equations
for a hyperbola, in Lemmas 8 and 9.

Lissajous figures: basics. Skip the part about the Tschebyshev
polynomials Tn and Sn.

3.1.4 Focus-directrix property, Theorems 3.1 and 3.2. Notably,
you should know how to carry out the algebraic calculations in the
middle of page 146.

3.1.5  Uniform parametrization of conics. Know it.

3.2.2  Cubic Bezier curves. Know the basic formula (3.13) and
Lemma 10. (Including the calculations we did to prove parts (1),
(2) and (5) of Lemma 10.)

3.2.3-4. The A-frame condition, and why it works. (Equations 3.14,
3.15 and 3.16, and what they mean.) Calculation
of Bezier control points from apical control points (i.e.
Equations (3.17-18)).

3.2.5  Interpolants. How one derives the matrix M-sub-(n-1) on
page 175, and how one evaluates its inverse.

3.2.6  Use of the B-spline language, pages 177-8.

3.3 The concepts described on page 184 -- general understanding

3.3.1-5 (Translations, rotations, reflections, glide reflections and
magnifications.) Know the formulas for these kinds of motions. Be
able to say which of the properties on page 184 are satisfied by any
of these motions. Theorem 3.16 on page 193 (statement only, not
proof).  Logarithmic spirals: the one-parameter group A-sub-theta
described on page 199. The spiral curves described on pages 199-200.