Geodesic Radius of Curvature Calculation Method

In summary, the geodesic (or tangent) radius of curvature of a geodesic circle can be computed using the formula \frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G}. To perform \partial G/\partial S, you will need to first express the functions in terms of arc length by re-parameterizing them. If you are working with a non-uniform rational b-splines surface and do not know the parametric equation of the geodesic path, you can use the formula \frac{d G}{d s}=G_u \frac{d u}{d s}+ G_v \frac{d v}{d s} to re-parameter
  • #1
manushanker20
3
0
I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula.

[tex]\frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G}[/tex]

where [tex]s[/tex] is the arc length parameter and [tex]E[/tex], [tex]G[/tex] are the coefficents of the first fundamental form.

Can you please tell me how to perfrom the [tex]\partial G/\partial S[/tex]? Since [tex]G=r_v\cdot r_v[/tex] I am not sure how to derivate it with respect to arc length

Thanks!
 
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  • #2
You will first express each of your functions in terms of the arc length - re-parameterize them.
 
  • #3
I am dealing with non-uniform rational b-splines surface and I don't know the parametric equation of the geodesic path. I just know a set of points on the geodesic then how to re-parameterize with arc length.

can I use [tex]\frac{d G}{d s}=G_u \frac{d u}{d s}+ G_v \frac{d v}{d s}[/tex]
 

What is the geodesic radius of curvature?

The geodesic radius of curvature is a mathematical concept used to measure the curvature of a curved surface at a specific point. It is defined as the radius of the circle that best approximates the curvature of the surface at that point.

How is the geodesic radius of curvature calculated?

The geodesic radius of curvature is calculated using the first and second fundamental forms of a surface, which contain information about the surface's shape and curvature. It involves taking the derivative of the surface's normal vector with respect to the surface's tangent vector at a given point.

What is the significance of the geodesic radius of curvature?

The geodesic radius of curvature is an important concept in geometry and differential geometry, as it helps us understand the shape and curvature of a surface. It is also used in fields such as physics and engineering, where curved surfaces are common.

How does the geodesic radius of curvature differ from other measures of curvature?

The geodesic radius of curvature is a specific measure of curvature that is used for curved surfaces in a three-dimensional space. Other measures of curvature, such as the Gaussian curvature and mean curvature, are used for different types of surfaces, such as curved lines and curved planes.

Can the geodesic radius of curvature be negative?

No, the geodesic radius of curvature cannot be negative. It is always a positive value, as it represents the radius of a circle, which cannot be negative. However, the sign of the geodesic radius of curvature can indicate whether the surface is concave or convex at a given point.

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