Mapping a stereographic projection to the inside of a sphere in ThreeJS

Question

When it comes to 3D animation, there are a lot of terms and concepts that I\'m not familiar with (maybe a secondary question to append to this one: wha

Answer 1

Working Example

I forked @manthrax's CodeSandbox.io solution and updated it with my own:

https://codesandbox.io/s/4w1njkrv9

The Solution

So after spending a day researching UV mapping to understand what it meant and how it worked, I was able to sit down and scratch out some trig to map points on a sphere to points on my stereographic image. It basically came down to the following:

1. Use arccosine of the Y coordinate to determine the magnitude of a polar coordinate on the stereographic image
2. Use the arctangent of the X and Z coordinates to determine the angle of the polar coordinate on the stereographic image
3. Use x = Rcos(theta), y = Rsin(theta) to compute the rectangular coordinates on the stereographic image

If time permits I may draw a quick image in Illustrator or something to explain the math, but it's standard trigonometry

I went a step further after this, because the camera I was using only has a 240 degree vertical viewing angle - which caused the image to get slightly distorted (especially near the ground). By subtracting the vertical viewing angle from 360 and dividing by two, you get an angle from the vertical within which no mapping should occur. Because the sphere is oriented along the Y axis, this angle maps to a particular Y coordinate - above which there's data, and below which there isn't.

1. Calculate this "minimum Y value"
2. For all points on the sphere:
   • If the point is above the minimum Y value, scale it linearly so that the first such value is counted as "0" and the top of the sphere is still counted as "1" for mapping purposes
   • If the point is below the minimum Y value, return nothing

Weird Caveats

For some reason the code I wrote flipped the image upside down. I don't know if I messed up on my trigonometry or if I messed up on my understanding of UV maps. Whatever the case, this was trivially fixed by flipping the sphere 180 degrees after mapping

As well, I don't know how to "return nothing" in the UV map, so instead I mapped all points below the minimum Y value to the corner of the image (which was black)

With a 240-degree viewing angle the space at the bottom of the sphere with no image data was sufficiently large (on my monitor) that I could see the black circle when looking directly ahead. I didn't like the visual appearance of this, so I plugged in 270 for the vertical FOV. this leads to minor distortion around the ground, but not as bad as when using 360.

The Code

Here's the code I wrote for updating the UV maps:

// Enter the vertical FOV for the camera here
var vFov = 270; // = 240;

var material = new THREE.MeshBasicMaterial( { map: texture, side: THREE.BackSide } );
var geometry = new THREE.SphereGeometry(0.5, 200, 200);

function updateUVs()
{
    var maxY = Math.cos(Math.PI * (360 - vFov) / 180 / 2);
    var faceVertexUvs = geometry.faceVertexUvs[0];
    // The sphere consists of many FACES
    for ( var i = 0; i < faceVertexUvs.length; i++ )
    {
        // For each face...
        var uvs = faceVertexUvs[i];
        var face = geometry.faces[i];
        // A face is a triangle (three vertices)
        for ( var j = 0; j < 3; j ++ )
        {
            // For each vertex...
            // x, y, and z refer to the point on the sphere in 3d space where this vertex resides
            var x = face.vertexNormals[j].x;
            var y = face.vertexNormals[j].y;
            var z = face.vertexNormals[j].z;

            // Because our stereograph goes from 0 to 1 but our vertical field of view cuts off our Y early
            var scaledY = (((y + 1) / (maxY + 1)) * 2) - 1;

            // uvs[j].x, uvs[j].y refer to a point on the 2d texture
            if (y < maxY)
            {
                var radius = Math.acos(1 - ((scaledY / 2) + 0.5)) / Math.PI;
                var angle = Math.atan2(x, z);

                uvs[j].x = (radius * Math.cos(angle)) + 0.5;
                uvs[j].y = (radius * Math.sin(angle)) + 0.5;
            } else {
                uvs[j].x = 0;
                uvs[j].y = 0;
            }
        }
    }
    // For whatever reason my UV mapping turned everything upside down
    // Rather than fix my math, I just replaced "minY" with "maxY" and
    // rotated the sphere 180 degrees
    geometry.rotateZ(Math.PI);
    geometry.uvsNeedUpdate = true;
}
updateUVs();

var mesh = new THREE.Mesh( geometry, material );

The Results

Now if you add this mesh to a scene everything looks perfect:

One Thing I Still Don't Understand

Right around the "hole" at the bottom of the sphere there's a multi-colored ring. It almost looks like a mirror of the sky. I don't know why this exists or how it got there. Could anyone shed light on this in the comments?