Determining a position vector-valued function for a parametrization of two parameters | Khan Academy | Summary and Q&A

TL;DR

Learn how to parameterize a torus using two variables, s and t, to define its position in three-dimensional space.

Key Insights

• 👾 Parameterizing a torus involves using two variables, s and t, to define its position in three-dimensional space.
• 🤐 The s parameter represents the rotation of a point around a circle, while the t parameter represents the rotation of the entire circle around the z-axis.
• 😵 The z-coordinate is straightforward to determine, while the x and y coordinates involve trigonometric functions and depend on both s and t.
• 🆘 The parameterization visualized in the s-t domain helps understand the varying shapes of the torus.
• 😃 The domain for the s and t parameters is typically defined as ranging from 0 to 2pi.
• 🧘 The position vector-valued function, r, can be used to calculate the exact position of any point on the torus given specific values of s and t.

Questions & Answers

Q: How are the s and t parameters used to parameterize a torus?

The s parameter represents the rotation of a point around a circle, while the t parameter represents how far the entire circle has rotated around the z-axis. Both parameters play a role in defining the position of the torus in three-dimensional space.

Q: What is the domain for the s and t parameters?

The s parameter ranges from 0 to 2pi, representing a complete rotation around the circle. The t parameter also ranges from 0 to 2pi, representing a complete rotation of the entire circle around the z-axis.

Q: How is the z-coordinate of the torus determined?

The z-coordinate is equal to a times the sine of the s parameter. This value represents the distance above the x-y plane for a given point on the torus.

Q: How are the x and y coordinates of the torus determined?

The x-coordinate is equal to the sine of the t parameter multiplied by (b + a times the cosine of the s parameter). The y-coordinate is equal to the cosine of the t parameter multiplied by (b + a times the cosine of the s parameter).

Summary & Key Takeaways

• To parameterize a torus, s and t are used as variables. The value of s represents the rotation of a point around a circle, while t represents how far the entire circle has rotated around the z-axis.

• The parameterization is visualized in the s-t domain, where s ranges from 0 to 2pi and t ranges from 0 to 2pi. The resulting shapes vary depending on the values of s and t.

• The z-coordinate of the torus is straightforward to define, as it is simply a times the sine of s. The x and y coordinates involve trigonometry and depend on both s and t.