# Surface area and radius of curvature

Hi All,

Imagine for a moment a mesh that starts out as a flat plane that never changes its total surface area.

This mesh is then laid down on a spherical surface and allowed to deform accordingly.

What I want to do is to be able to control the radius of the sphere on which the plane rests - in other words - control the radius of curvature of the plane simply by adjusting the radius of curvature, and have the plane mesh adjust its curvature accordingly. The end result would be that as a radius increases, the mesh becomes more flat. If the radius decreases, the mesh curvature becomes more spherical.

I’ve tried this with warping, which changes the plane’s area; nerbs, although it looks promising, has the same problem, I think.

I’m starting to think that using a cloth is my only solution, but I’d like to know if there is something easier to avoid simulating if possible? Python scripts, perhaps?

Any help much appreciated,

Henry

Okay, let’s see if this is what you want. Start with a cube, subdivide it several times. Select to sphere @ 100% Then switch over to solid view with occlude background geometry selected. Go to face select mode and select the faces you want for your curved plane. Now select inverse and delete vertices. This should leave you with a curved plane that is scalable. Set smooth and there ya go.

Hope that helps.

Will try,

Thanks!

It looks like what you suggested is almost what I’m looking for.

What is left is for me to be able to scale the radius of curvature only, while the surface area of the faces of the mesh remains the same.

I hope that clarifies it a bit.

Thanks for helping!

Henry

Well, it won’t be as simple as you want it, but it’s just a few clicks longer.

1. Add a metaball under the subdivided plane
2. Scale it to be as curved as you want (edit mode)
3. Go to edit mode of the grid, turn on retopo and press retopo all. Be sure you’ve got at least one 3Dview in top view. With multiple 3D windows you have to click the one in the top view.
4. To change the radius, just scale the ball and retopo again!

That’s the one.

Thanks a mil,

Henry

Bending a planar grid while keeping its surface area constant is not that easy as it looks.
Retopology is not a solution for this problem because of the way it projects the vertices.
Probably python script could do the math. Here is the formula for the surface area of a spherical polygon
http://mathworld.wolfram.com/SphericalPolygon.html

Thanks Syziph,

I figured that retopology would be my best solution short of writing a custom script.

Using retopology with some other tools in the mix might get the effect I need.

This touches on what was written in one of the Blender books that Blender - like other animation software - strives for control of the visual effect, rather than the underlying physical accuracy.

But I see as Blender becoming both - eventually, considering its price tag…

I agree!
It is not always nescessary to work with great accuracy.
Besides that it is not possible to create a tool for every modeling problem