This is the webpage for the course *Geometry and Algorithms in Three-Dimensional Modeling* at Oklahoma State University.

Classes will consist of lectures on the mathematical underpinnings of 3D design software, and "workshop classes", spent learning the use of the software and working on projects.

### Lectures

We will cover a number of sections from the course text, *Applied Geometry for Computer Graphics and CAD* (Springer Undergraduate Mathematics Series) 2nd Edition, by Duncan Marsh. (The text is freely available in electronic form from library.okstate.edu.) We will go through parts of chapters 1-3 (transformations, homogeneous coordinates), and chapters 6-9 (Bézier curves, B-splines and surfaces).

### Homeworks

Homeworks will consist of standard problem sets to be handed in in class, or virtual 3D models/animations/interactive content to be handed in via Canvas.

### Tutorials

One cannot effectively learn how to use a piece of software by listening to lectures. Rather, one should use the software. The modelling homework sets and projects will give concrete goals to guide exploration of the software. Students will also follow tutorials.

### Midterms

There will be two take home midterms. There will be no final exam.

### Projects

There will be three major projects due throughout the semester. The following schedule and topics are tentative.

#### Project 1: Goblet

Design a goblet. This will likely be a surface of revolution (although if you wish you may depart from this). The goblet should hold *precisely* 100ml of water. The shape should be interesting in some way, either mathematically, culturally, and/or aesthetically. Along with the 3D file, students will complete a writeup on the mathematics, *including a calculation of the volume of the goblet*, and other relevant details of the design and present their work to the class.

- Project proposal due: Tuesday
**February 9**. Before class, have decided on, and sketched the rough shape of the goblet, and have thought about how to create the shape. In class we will meet to discuss the project. - Prepare project ideas for Thursday
**February 11**. Before class, have researched the ideas or functions you will need to design the goblet, and have started their writeup. In class we will work on the 3D design and writeup. - Product due date and draft writeup due: Tuesday
**February 16**. In class we will work on the writeup and presentation. - Completed writeup due, and presentations: Thursday
**February 18**.

#### Project 2: Programming project

Produce an artwork or illustration, in the form of a 3D design, an animation, or an interactive demonstration, using some form of programming, in either Rhino-Python or Grasshopper. Some possible ideas: Show a parametrised curve in space (a graph, the path of an object in motion etc.), show the graph of a function of two variables, make a tree-like fractal (or perhaps investigate L-systems), simulate a natural process (nautilus shell, pinecone etc.).

- Project proposal due: Thursday
**March 16**. Before class, have decided on what to design, and have thought about how to create it. In class we will meet to discuss the project. - Product due date and draft writeup due: Thursday
**March 23**. In class we will work on the writeup and presentation. - Completed writeup due, and presentations: Tuesday
**March 25**.

#### Project 3: Illustrate something mathematical

Produce a model (or animaion, or interactive demonstration) illustrating a mathematical object, process, algorithm etc. The model may be drawn from your previous mathematics classes, or (even better) mathematics that is new to you. I can give you references to read. For example there are many ideas in symmetry or knot theory or other areas of mathematics that are accessible and would be good to illustrate with a 3D model.

- Project proposal due: Tuesday
**April 8**. Before class, have decided on what to design, and have thought about how to create it. In class we will meet to discuss the project. - Products and writeups due, presentations: Tuesday
**April 27**and Thursday**April 29**.

#### The "make something cool" alternative

For any of the three graded projects, you can come to me**before**the initial "project proposal due" date, and suggest doing some entirely different project, drawn from your personal interests, other classes, or anything else you want to make. If the project has a suitable amount of mathematical content, then you may replace the graded project with your own.

### Resources

- Rhinoceros the 3D design software we will use for the class. The software is available for Windows and Mac through the College of Arts and Sciences (details to come). Meanwhile, a 90 day free trial of the software is available from here.
- Grasshopper, a visual scripting language built in to Rhinoceros.
- Rhino-Python, an implementation of the programming language Python within Rhinoceros.
*Essential Mathematics for computational design*- Third Edition, by Rajaa Issa. This very briefly covers the mathematics used in Rhinoceros.- Meshmixer can help with unioning together meshes that Rhinoceros has trouble with (using Edit -> Make Solid).
- Laura Taalman: 365 mathematical 3D printing projects, and even more projects.
- Chris Hanusa: Math with Mathematica course.
- Oliver Knill and Elizabeth Slavkovsky: Thesis project on 3D printing in education.
- Thingiverse (3d file repository).
- More inspiration: David Bachman, Ken Baker, Vladimir Bulatov, Bathsheba Grossman, George Hart, Oliver Labs, Henry Segerman, Nervous System, Carlo Séquin, Oskar van Deventer.

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