Constructing a Torus: Logical Justification

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In summary, we are trying to create a torus with a given axis of symmetry and radius of the tube. The equation (R-√(x²+y²))²+z²=r² describes all the points in ℝ³ that are a distance of r from the circle {(x,y)∈ℝ²:x²+y²=R²}. By substituting t=R-√(x²+y²) into the equation t²+z²=r², we get the equation for the torus. The justification for this is that we are essentially rotating the smaller circle t²+z²=r² around the larger circle x²+y²=R² to create a three-dimensional surface. This is why we can use a
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Wendel
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Homework Statement


Suppose we wish to construct a torus {(x,y,z)∈ℝ³:(R-√(x²+y²))²+z²=r²} whose axis of symmetry is the z-axis and the distance from the center of the tube to the center of the torus is R. Let r be the radius of the tube.

Homework Equations


The large circle in the xy-plane: x²+y²=R².
The smaller circle which revolves around the larger one: t²+z²=r² where t is the component of the distance from the point on the torus to the center of the tube that lies in the xy-plane.

The Attempt at a Solution


(R-√(x²+y²))²+z²=r² consists of all the points in ℝ³ a distance of r from the circle {(x,y)∈ℝ²:x²+y²=R²}. And t=R-√(x²+y²). Substituting this in the place of t in the equation t²+z²=r² gives us the equation of the torus. My question is why does this work; What is the logical justification for doing so? I don't understand the exact process by which we go from a one-dimensional object in a plane, to a surface in ℝ³.
 
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You have one constraint equation for a three-dimensional space. This describes a two-dimensional surface. I do not understand your reference to a one-dimensional object in the plane.
 

FAQ: Constructing a Torus: Logical Justification

1. What is a torus and why is it important to construct one?

A torus is a three-dimensional geometric shape with a circular cross section and a hole in the center. It is important to construct a torus because it is a common shape found in nature and has many applications in mathematics, physics, and engineering.

2. What is the logical justification for constructing a torus?

The logical justification for constructing a torus is to understand its properties and relationships with other geometric shapes. By constructing a torus, we can gain insights into its curvature, volume, and surface area, which can be applied to real-world problems.

3. What are the steps involved in constructing a torus?

The steps involved in constructing a torus include drawing two circles of different radii, connecting the points on the smaller circle to the larger circle, rotating the smaller circle around the larger circle, and filling in the space between the circles to create a solid torus shape.

4. What tools or materials are needed to construct a torus?

To construct a torus, you will need a compass to draw circles of different radii, a straight edge to connect points on the circles, and a protractor to measure angles. Additionally, you may need paper, a pencil, and a ruler to draw and measure the torus.

5. What are some real-world applications of a torus?

A torus has many real-world applications, including in architecture, where it is used in the design of buildings and structures. In engineering, a torus is used in the design of gears, pipes, and other circular objects. It also has applications in physics, such as in the study of magnetic fields and fluid dynamics.

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