Engineering a body with an intrinsic curvature

[geordief]

I don't know much about differential geometry but I hope this is a good place to ask and that my question "makes sense"

I have heard that an ornamental cabbage leaf is an example of a surface with an intrinsic curvature.

If one wanted to make such a surface from scratch(and to detailed specifications) how would one go about it?

Again ,if one wanted to describe a particular "ornamental cabbage leaf" surface mathematically how could this be done?

And , thirdly ,I suppose is there a general algorithm or a mathematical method to create or describe surfaces that have intrinsic curvature?

 

[andrewkirk]

Any contiguous, nonzero area on the surface of a sphere is a surface with intrinsic curvature. I imagine that is easier to engineer than a cabbage leaf.

I don't know about ornamental cabbage leaves but, since ordinary cabbage leaves are shaped around a large part of a sphere, they would also have intrinsic curvature (otherwise they wouldn't fit without wrinkling). That includes the ordinary, non-wrinkly green and red cabbages. There's no need to go for the wrinkly Savoy cabbages. Their leaves have intrinsic curvature as well, but it would be more complex to make a replica.

 

[geordief]

Thanks.Would the only way to replicate the surface be to apply differential equations at various points on a sphere (ball?) and build up the shape gradually?

There is no formula that can be fed into ,say a 3rd printer

 

[Nidum]

For practical rather than theoretical purposes you just model your surface/solid using a 3DCAD system . Files for 3D printing can then be generated automatically .

3D CAD systems are available from children's toy level up to very sophisticated professional level .

This is a fun one to play with and it's free :

https://www.tinkercad.com/

 

[Nidum]

https://www.tinkercad.com/things/?page=7

Not quite a cabbage but the frog is good .

 

[geordief]

Thanks. I am actually interested in theoretical rather than practical.

I am wondering how you can (ideally with one formula or else series of formulas) create a model of an intrinsic surface (be it ever so elementary)

If I want to create a model of a sphere ,I think it goes along the lines of x^2 +y^2 +z^2 - constant =0 (hope that is right)

That is ,I think a surface with intrinsic curvature.

Perhaps I have answered my question.

Can that sphere (is it a ball) be modeled using differential equations rather than the formula I have just used?

 

[Nidum]

Theoretical definition is just going to be based on a mesh of nodes and individual element properties . Same as FE and probably same as some older types of 3D CAD internally .

 

[geordief]

So how does a plant grow a leaf to a particular shape? Do we know the mathematical formula(or formulae?) behind it

 

[Nidum]

https://www.google.co.uk/search?q=f...-8&oe=utf-8&gws_rd=cr&ei=AtskVsHLA8rtUuf3kbAG

 

[geordief]

Thanks(that should have been obvious to me ). I wonder if it is possible to generate the spherical surface using a fractal formula...

 

[Nidum]

I think that you could probably trace back to the underlying maths from here :
[PLAIN]http://mandelbulb.com/2014/mandelbulb-3d-mb3d-fractal-rendering-software/[/PLAIN]
http://mandelbulb.com/2014/mandelbulb-3d-mb3d-fractal-rendering-software/

In any case excellent software and certainly demonstrates that 3d curved surfaces can be generated quite easily .

 

[zinq]

I confess to being puzzled to read a statement that some specific surface "has intrinsic curvature". Every surface has intrinsic curvature! (OK, just surfaces that are twice continuously differentiable.)

By a "surface" here I mean a metric surface, with the natural type of metric that surfaces and any manifolds can have: a Riemannian metric: https://en.wikipedia.org/wiki/Riemannian_manifold. (This is a smooth choice of inner product on the space of tangent vectors at each point of the surface. Using this, any smooth curve on the surface has a well-defined length, namely the integral of the lengths of its velocity vectors.)

Intrinsic curvature just means the curvature that can be measured by referring only to the geometry of the surface itself, and not to a space (like Euclidean space) in which the surface may happen to find itself.

In fact, every surface (and I'm thinking of any two surfaces as being the same if there is a distance-preserving bijection between them — an isometry) can in principle be defined without reference to any space containing it. For example, a square torus T² can be defined as

T² = {(x,y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} / ~​

where ~ indicates that, for all y, (0,y) and (1,y) are to be considered as the same point, and likewise for all x, (x,0) and (x,1) are to be considered the same point. This can be thought of as the cartesian product of two circles, each of circumference = 1.

For convenience, let m(x₁,x₂) be defined as the distance function on a circle of circumference = 1 in terms of two points x₁, x₂ with 0 ≤ x₁ ≤ 1, 0 ≤ x₂ ≤ 1. That is,

m(x₁,x₂) = min{|x₂-x₁|, 1-|x₂-x₁|}.
​

(The distance between two points on a circle is the shorter of the two arcs between them.)

Then T² can be given its natural distance formula

D((x₁,y₁), (x₂,y₂)) = √( m(x₁,x₂)² + m(y₁,y₂)² ).
​

Note that we haven't made any assumption about "where" this surface T² is supposed to be. It just is. (Incidentally, this surface does not exist in 3-space, but it is easy to construct it in 4-space.)

P.S. Incidentally, one way to approximate a surface of constant negative curvature in R³ is to put together regular planar heptagons, 3 per vertex. To make this a smooth surface, negatively curved heptagons could be used. Interestingly, one can continue this process only so far before the surface bumps into itself! There is no way to place a smooth surface of constant negative curvature in R³, extended indefinitely in all directions, without its bumping into itself. This is a theorem of Hilbert: https://en.wikipedia.org/wiki/Hilbert%27s_theorem_(differential_geometry ). This may explain why a Savoy cabbage is always finite (:-)>.

 

[andrewkirk]

I confess to being puzzled to read a statement that some specific surface "has intrinsic curvature". Every surface has intrinsic curvature! (OK, just surfaces that are twice continuously differentiable.)

Saying a surface 'has intrinsic curvature' is an abbreviation for 'has nonzero intrinsic curvature'. Yes, every twice differentiable pseudo-Riemannian manifold has intrinsic curvature, but the curvature may be everywhere zero, as in the case of Euclidean space.

 

[zinq]

Okay. But is there some distinction between "intrinsic" curvature and "extrinsic" curvature (or just plain curvature)?

 

[andrewkirk]

Yes. A cylinder has extrinsic curvature but no intrinsic curvature. Extrinsic curvature is a feature of an embedding. Intrinsic curvature depends only on the metric of the manifold.

 

[zinq]

But if we are talking throughout about Gaussian curvature (as we in fact are), then Gauss's Theorem Egregium shows that no matter how a surface is embedded in 3-space, its extrinsic Gaussian curvature does not change (and is always equal to its intrinsic Gaussian curvature).

(By "extrinsic" Gaussian curvature I mean the product of the principal curvatures at any point of the surface.)

 

[andrewkirk]

Ah, OK. Then we've been talking about different things. Crossed wires.