Parametrization and Visualization of Quadric Surfaces

Sayonita Ghosh Hajra, California State University-Sacramento, Mathematics and Statistics
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This activity is exploratory and collaborative MATLAB activity. Students will explore and visualize graphs of quadric surfaces. In this activity, students will parametrize equations of surfaces and explore families of various quadric surfaces.

Learning Goals

Students will learn about quadric surfaces and their parametrization. MATLAB will be used to plot these surfaces. Using MATLAB students will visualize the quadric surfaces. Students will develop conceptual understanding and critical thinking by examining equations of various quadric surfaces, students can imagine how a quadric surface changes when the parameters in the equation are changed.

Context for Use

California State University, Sacramento is a public university in Sacramento. This activity is intended for a multivariable calculus course. The course is a continuation of Calculus II for students majoring in mathematics, physical science and engineering. The class size is 30 and is offered 50 minutes for four days a week. This activity is intended for two 50-minute class periods. For this activity, students are expected to be familiar with the MATLAB interface, mathematical operations, functions, and 3D plotting. This activity is for week 4-5; this activity could engage students more productively, ideally after students have seen polar, cylindrical and spherical coordinates. This activity goes well with the introduction of quadric surfaces and can be used prior to the discussion of vector-valued functions.

Description and Teaching Materials

This activity can be used after discussing the equation of the quadric surfaces and before the section on vector-valued functions. This activity can be either conducted in a class with personal laptops or in a lab. Students will work in pairs for this activity. I chose MATLAB as it has a good collection of toolboxes, it is available for free to all students in my school and it is highly likely that students have used MATLAB in other courses as students are mainly engineering and computer science major.

Class_Activity.pdf (Acrobat (PDF) 1.4MB Nov7 19)

Student Worksheet.pdf (Acrobat (PDF) 530kB Nov7 19)

Class_Activity.mlx (MATLAB Live Script 3MB Nov7 19)

Teaching Notes and Tips

This activity serves better after the discussion of spherical coordinates and goes well with the introduction of quadric surfaces. It is useful to form groups; groups of two is ideal but could be modified for three as well.


This activity is assessed through a form of formative assessment. This aligns with the learning objectives for the class activity to assess whether students (individually and collectively) are learning and provides an opportunity for the instructor to check students' understanding of prior knowledge of polar, cylindrical and spherical coordinates.

First students will write parametrizations for Ellipsoid, Elliptic Paraboloid, Hyperbolic Paraboloid, and Cone in Activity 1. Solution keys will be provided so that students can check their answers. It is highly important that students understand parametrization before visualizing the 3D-shapes in MATLAB. Students will be asked to correct their work based on the solution key and the activity sheets will be collected by the instructor. Students will receive full points for the Activity 1 for turning in their corrected work.For the Activity- 2, 3 and 4, students will be graded based on the following rubric. (File attached).

Rubric.pdf (Acrobat (PDF) 58kB Nov7 19)

References and Resources

Calculus: Early Trans, Author: Stewart, Publisher: Cengage Learning, Edition: 8th, Year Published: 2015

Multivariable Calculus: Early Transcendentals, Author: Rogawski & Adams, Publisher: W.H. Freeman & Company, Edition: 3rd, Year Published: 2015