Finding the Centroid of a Spherical Crescent

In summary, the conversation discusses the topic of dealing with spherical "crescents" and incomplete annuli and finding the centroid of a partially intersecting spherical crescent. The individual has found the area for this case through some research, but is struggling with finding more information. They suggest using geometric approaches or integrating over a spherical cap to find the center of mass. They also mention using cylindrical coordinates with the center of both spheres on the z axis and having two remaining integrals over z and roh. Various resources are provided for further reading on the topic.
  • #1
Dehstil
1
0
Hello,

What I really want to do is deal with spherical "crescents" and incomplete annuli and see how well they are approximated by spherical caps, but here is my question:

How would you go about finding the centroid of a spherical crescent (one spherical cap minus the other) in the case when they are partially intersecting?

After some digging, I've managed to find the area but not much else for this case:
Page 10 on: ati.amd.com/developer/siggraph06/Oat-AmbientApetureLighting.pdf[/URL]
Page 2: [url]www.cse.ust.hk/~psander/docs/aperture.pdf[/url]
Page 12: [url]www3.interscience.wiley.com/cgi-bin/fulltext/121601807/PDFSTART[/url]

I've attempted some geometric approaches but have not gotten very far. Perhaps knowing how to integrate over a spherical cap or the intersection of two spherical caps would be useful in a calculus-based "center of mass" approach.
 
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  • #2
I'm pretty sure you want cylindrical coordinates with the center of both spheres on the z axis. Integral over theta is easy via symetry. You would have two remaining integrals over z and roh. You need to write equations for the surface of each sphere in terms of z and roh.
 

What is a spherical crescent?

A spherical crescent is a geometric shape that is created by taking a sphere and cutting out a smaller, spherical section from it. This creates a crescent-shaped region on the surface of the sphere.

Why is finding the centroid of a spherical crescent important?

The centroid is the center of mass of an object or shape. For a spherical crescent, finding the centroid is important for understanding the distribution of mass and how the shape will behave when under external forces.

How is the centroid of a spherical crescent calculated?

The centroid of a spherical crescent can be calculated using the formula:
C = (2Rsinθ - 2Rsin(θ/2) + Rθ) / (3θ), where R is the radius of the sphere and θ is the angle of the crescent.

Can the centroid of a spherical crescent be located outside of the shape?

Yes, it is possible for the centroid of a spherical crescent to be located outside of the shape. This can occur if the crescent is not symmetrical or if the angle θ is very small.

What are some real-world applications of finding the centroid of a spherical crescent?

The concept of finding the centroid of a shape is used in various fields such as engineering, architecture, and physics. For example, in structural engineering, the centroid of a shape is used to determine the center of gravity and the distribution of forces in a structure. In architecture, it is used to ensure the stability and balance of a building design. In physics, the centroid is used to calculate the moment of inertia of an object, which is important for understanding its rotational motion.

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