Parameterizing the Surface of a Torus

Using two parameters, parametrize the surface of a torus (doughnut shape) in three dimensions, considering a circle of radius $a$ at a distance $b$ from the center (z-axis), and varying the parameters to map the surface of the torus.

In this problem, we delve into the parameterization of a torus, which is a fundamental shape in vector calculus and three-dimensional geometry. Parameterizing surfaces involves the use of mathematical expressions to map points from a two-dimensional domain into three-dimensional space, effectively describing the entire surface collectively. For a torus, which can be visualized as the surface of a doughnut, the problem reduces to understanding how to map a circle that revolves around an axis to create this familiar shape.

When parameterizing a torus using two parameters, typically denoted as u and v, it is crucial to utilize trigonometric functions to capture the circular symmetry of the shape. The parameter u can be thought of as controlling the position along the major circle, while v varies around the minor circle. Through trigonometric identities, each combination of these parameter values can be uniquely translated into a point in three-dimensional space.

Understanding the underlying geometry of a torus is essential for solving this problem and involves grasping concepts of distance from an axis (often defined by radius '$b$') and the radius of revolutions ('$a$' for the minor circle). This exercise reinforces the principles of three-dimensional transformations and aids in visualizing complex shapes, which are essential skills in fields like computer graphics, mechanical design, and theoretical physics.

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