Curve Inside a Sphere: Differentiating Alpha

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Thread starter Celso
Start date Sep 11, 2019
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Calculus Curve Differential geometry Geometry Sphere

In summary, a curve inside a sphere is a mathematical concept where a curve is drawn on the surface of a sphere, often using the shortest possible path. It is different from a regular curve as it follows the curvature of the sphere. Differentiating alpha in a curve inside a sphere is important in calculating the curve's curvature and has practical applications in fields such as geodesy, navigation, and computer graphics.

Sep 11, 2019

Celso

TL;DR Summary: Let ##S^2 \subset R^3## the sphere whose center is at the origin and has radius 1.There is a function ##\alpha \colon I \rightarrow R^3## parameterized by arc length, regular, such as ##\alpha (I) \subset S^2## with constant and positive curvature.

Proof that there exists a differentiable function ## g \colon I \rightarrow R## such as ##\alpha (s) = - \frac{1}{k} \vec{n} + g(s) \vec{b}##.

Show that ##(\frac{1}{k})^2 + (g(s))^2 = 1##, therefore ##g(s)## is constant

Honestly I don't know where to begin. I started differentiating alpha trying to show that its absolute value is constant, but the equation got complicated and didn't seem right.

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Sep 14, 2019

fresh_42

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I think the constant curvature of ##\alpha## is the key here. What does the curvature formula say?

1. What is a curve inside a sphere?

A curve inside a sphere is a mathematical concept in which a curve is drawn on the surface of a sphere. This curve can be any shape or size and is often used to represent real-world objects such as a circle of latitude or a meridian on a globe.

2. What is the purpose of differentiating alpha in a curve inside a sphere?

Differentiating alpha in a curve inside a sphere is used to find the rate of change of the curve at a specific point. This can help us understand the behavior of the curve and make predictions about its future movements.

3. How is alpha differentiated in a curve inside a sphere?

Alpha is differentiated in a curve inside a sphere using the chain rule from calculus. This involves taking the derivative of the curve with respect to the sphere's surface and then multiplying it by the derivative of the sphere's surface with respect to alpha.

4. What is the significance of the curve's curvature in a sphere?

The curvature of a curve inside a sphere is important because it tells us how much the curve is changing direction at a given point. A higher curvature means the curve is curving more sharply, while a lower curvature means the curve is relatively flat. This can help us understand the shape and behavior of the curve.

5. How is the curve inside a sphere related to real-world applications?

The concept of a curve inside a sphere has many real-world applications, such as in geography, navigation, and astronomy. For example, curves on the Earth's surface can be represented as curves inside a sphere, which can help us understand and navigate our planet. In astronomy, curves inside a celestial sphere can represent the paths of stars and planets in the sky.