Note to instructor: Need picture here. Either spring 98 result, or Put Kirk's donut picture in top corner. Use columnspan 2

 MAT 272 Calculus and Analytic Geometry III Spring 1998 Project 1 Parameterized curves M.Kawski

Due dates: (subject to negotiation on Monday Jan 26)

 Friday, January 30 postponed toMonday February 2 Time table for team meetings, action plans, team-roles (rotate these), and responsibilities. Sketch of word on graph paper with clearly labeled axes, labeled tickmarks, and marked times at selected points Friday, February 6 All technical work completed (except possibly for curvature and reparameterization). 1 page sample of formulas and MAPLE code Sketch/outline for 1-page poster (hand-drawn outline is fine). Time-table for team meetings, action plans, team-roles and responsibilities. Friday, February 13 Return to the top. Final write-up and poster (or WWW-page) due.

• 1. Fix a word, number or name.
The only restrictions are that it must have at least four characters, and must include at least one horizontal, vertical, slanted, and curved segment each. Think of programming a pen- plotter, or a welding-robot (constant speed !!).
• Consider the word as a curve in the plane and find a single parameterization for the entire curve:
At any time the "robot/plotter" can be only at one location and hence different pieces must utilize distinct time intervals.
Note: use equal scales on the coordinate axes (angles are important in the velocity/acceleration parts of this project).
• Use MAPLE to plot the parameterized curve.
Visit this worksheet for help for an effective MAPLE syntax.
Use the plot option scaling=constrained.
• By hand mark typical velocity and acceleration vectors at various points along the curve (on print-out).
• Calculate and graph the speed (as a function of time).
• Calculate and graph the perpendicular component of the acceleration vectors (as a function of time).
• Calculate and graph the curvature (as a function of time).
• Try to reparameterize the curve by arc length (i.e. re-scale the parameterization to achieve constant speed equal to one).
Then repeat the last four steps.
• Prepare a laminated 11"x17" poster that summarizes your work. As possible audiences think of an accreditation or curriculum committee as a possible audience, your prospective employer, or other students who you want to impress with the work performed in this class. See the advice below.
• For bonus credit:
Pick your favorite parameterized surface in 3-space.
Consider your word as written in the parameter-plane (e.g. what you called x and y above, simply think of as theta and phi, or u and v).
Use a computer to plot the word spelled on the surface in 3-space.
In this part the artistic and practical components will earn the most credit.

Deliverables:
The final write-up must be signed by all team-members. It should include a brief narrative about the project; complete technical work (formulas, calculations, justifications and explanations), but put long computer printouts into appendix, the plot (computer-printouts) of the curve with marked vectors as above; a plot of the surface (optional, color printer available).

Evaluation:

Incomplete work will be rejected and assigned 0 points. Intermediate time-lines are worth 20 points each. Final write-up: 40 points for technical correctness. 20 points for presentation including the laminated poster. The WWW page may earn 10 bonus points but only if final write-up + intermediate timelines are over 90 pts. The curve on a surface may earn up to 20 bonus points!
Unless negotiated otherwise each team will get only one grade. However, if all students in a team agree, they may choose to allocate a small portion to reward special efforts etc. Guidelines: In each team the average must agree with the grade assigned to the team. Each team member's credit must lie between 80% and 120% of the team average. In case the team chooses not to give every member the same number of points, they must turn in a separate page together with the final report, that is signed by all team members, and that specifies what percentage each team member shall earn.