# Creating a UV Map for a 3D Cone

I have the following problem I am currently facing.

I am creating the vertices for a cone based on the number of segments and the radius of the base and top.

This is done as follows:

#include <cmath>
#include <fstream>
#include <numbers>
#include <iostream>
#include <fmt/core.h>

struct Vec3D
{
double x, y, z;
Vec3D(double x, double y, double z) : x(x), y(y), z(z) {}
};

int main()
{
int n = 4;       // number of segments
double r1 = 0.8; // radius of the base, assume meters
double r2 = 1.0; // radius of the top, assume meters
double h = 1.0;  // height of the cone, assume meters

auto get_coordinates = [&n](int32_t vertex_id, double radius, double height) {
double theta = vertex_id * 2 * std::numbers::pi / n;
double x = radius * cos(theta);
double y = radius * sin(theta);
double z = height;

return Vec3D(x, y, z);
};

for (int i = 0; i < n; i++)
{
Vec3D pt = get_coordinates(i, r1, 0.0);
std::cout << fmt::format("v {} {} {}\n", pt.x, pt.y, pt.z);
pt = get_coordinates(i, r1, h);
std::cout << fmt::format("v {} {} {}\n", pt.x, pt.y, pt.z);
}
}


(godbolt: https://godbolt.org/z/njdxznj9c)

Which works perfectly fine, but now I want to automatically create a UV map, which of course in a cylinder case can be very straight forward by just distributing the uv coordinates for the top vertices between [0, 1] for u and v = 1.0, and for the bottom the same with v = 0.0.

Now in a cone case this would lead to texture distorting.. which I want to avoid and therfore find an arc-like mapping, such that if the printed texture has the correct ratios and sizes (to the actual cup) you can wrap it without distortion, basically the same as blender can do Assuming now only calculating the following points a, b, c, d my plan was that those points should be possible to calculate and from there on use some sort-of arc parametrisation based on the radii to calculate the other points. points a and b can be calculated under the assumptions of the coordinates that it is in relation to the ratio of the 2 cone radii.

But I am not really sure how to follow up on that. I'm glad for any help on that. 