generative modeling for Rhino
If the mathematical definition of a geodesic curve is: the shortest path between two points on a surface, why the geodesic component returns a curve like this? that's a longer curve compared with others possible paths...
A geodesic is not actually defined as the shortest path between two points on a surface - it is only locally the shortest. So for any 2 points close together on the path it is the shortest route between them, but the whole path may not be the shortest one between its ends.
For example, you can draw a great circle passing through any 2 points on a sphere. If the 2 points are not opposite each other on the sphere there will be a long way round and a short way between the points along the circle, but both are geodesic.
On a surface that you could make out of paper without stretching it (a developable surface), such as a cylinder, geodesics are lines which would be straight if you unrolled the paper. Between any 2 points on the cylinder these geodesics will follow a helix, but there are infinitely many possible ones to choose from, depending on how many times you wrap around the cylinder.
Minimal surfaces are similar - they do not necessarily minimize the total area of the surface spanning the given boundary curves, but each little piece of the surface is the minimum area surface for its own small boundary (and for a given boundary there are sometimes multiple possible solutions).
I think that your response is addressing the issue of a geodesic on a closed surface. Giancarlo's example is an open surface so it won't have a closed curve for a geodesic at any time. I wonder if his surface has some anomaly that is causing the result he is getting.
"Although defining a geodesic as the shortest arc between two points on a surface gives the main idea of a geodesic, there is a problem with it as a definition. Not every geodesic is a shortest path in the large, as can be seen by noting that on the surface of a sphere every arc of a great circle is a geodesic even though an arc will be the shortest path between two points only if that arc is not greater than a semicircle. From this example we see that a geodesic can be a closed curve. Because of this difficulty a geodesic is often defined as an arc C on a surface S at each point of which the principal normal coincides with the normal to S — or an arc at every point of which the geodesic curvature vanishes identically." http://www.solitaryroad.com/c335.html
So the bias 'short path' is just an unfortunate consequence of the popularization of the 'geodesic' dome and the case of the sphere or sphere-like surface.
Create a mesh for your surface and try Giulio's Shortest Walk routine.
how is approximate geodesic component in grasshopper?
Theoretically, the surface normal and curve(geodesic) main normal must match, isn´t it?
i´m trying to probate this as the result is in the image,
Here's a link that might help.
The circle on the cone in your definition is not a geodesic as a circle will not unroll into a straight line but rather an arc or on a cone of zero height a circle.