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

Classes will take place on Tuesdays and Thursdays, 10:30 - 11:45am, in MSCS 514, and 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. We will also use the 3D printing lab, MSCS 421. Students in the class will have 24 hour keycard access to the lab with their university ID card.

### 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 to be handed in via D2L, or real-life 3D printed models to be handed in in class.

### 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.

- Rhino tutorials.
- Makerbot Replicator instruction manual.
- Preparing files for 3D printing with Rhino.

### 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 and 3D print 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 print, students will complete a two page 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 5**. 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 7**. 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. - 3D print due date and draft writeup due: Tuesday
**February 12**. In class we will work on the writeup and presentation. - Completed writeup due, and presentations: Thursday
**February 14**.

#### Project 2: Programming project

Produce a model (an artwork, an illustration, etc.) 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 14**. 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. - Prepare project ideas for Tuesday
**March 26**. Before class, have researched the algorithms you will need to make your design, and have started the writeup. In class we will work on the code, 3D design and writeup. - 3D print due date and draft writeup due: Thursday
**March 28**. In class we will work on the writeup and presentation. - Completed writeup due, and presentations: Tuesday
**April 2**.

#### Project 3: Illustrate something mathematical

Produce a model 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 printed model.

- Project proposal due: Tuesday
**April 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. - Prepare project ideas for Thursday
**April 18**. Before class, have researched the mathematics you will illustrate in your design, and have started the writeup. In class we will work on the 3D design and writeup. - 3D prints and writeups due, presentations: Tuesday
**April 30**and Thursday**May 2**.

#### 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 loaded onto the laptops we will use in class, and the computers in the 3D printing lab. A 90 day free trial of the software is available from the website, and although it won't be completely necessary for the class, I encourage you to purchase a copy for your own computer. With student credentials, Rhino is currently available for Windows and Mac for $195.
- 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).
- 3D printing course by Stratasys. Project ideas, sample files.
- 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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