Develop Reference

Random Rotation on a 3D sphere given an angle

This question is in between computer graphic, probability, and programming, but since I am coding it for an Unity project in C# I decided to post it here. Sorry if not appropriate.
I need to solve this problem: given a object on a 3d sphere at a certain position, and given a range of degrees, sample points on the sphere uniformly within the given range.
For example:
Left picture: the cube represents the center of the sphere, the green sphere is the starting position. I want to uniformly cover all surface of the circle within a certain degree, for example from -90 to 90 degrees around the green sphere. My approach (right picture) doesn't work as it over-samples points that are close to the starting position.
My sampler:
Vector3 getRandomEulerAngles(float min, float max)
{
float degree = Random.Range(min, max);
return degree * Vector3.Normalize(new Vector3(Random.Range(min, max), Random.Range(min, max), Random.Range(min, max)));
}
and for covering the top half of the sphere I would call getRandomEulerAngles(-90, 90).
Any idea?

Try this:
public class Sphere : MonoBehaviour
{
public float Radius = 10f;
public float Angle = 90f;
private void Start()
{
for (int i = 0; i < 10000; i++)
{
var randomPosition = GetRandomPosition(Angle, Radius);
Debug.DrawLine(transform.position, randomPosition, Color.green, 100f);
}
}
private Vector3 GetRandomPosition(float angle, float radius)
{
var rotationX = Quaternion.AngleAxis(Random.Range(-angle, angle), transform.right);
var rotationZ = Quaternion.AngleAxis(Random.Range(-angle, angle), transform.forward);
var position = rotationZ * rotationX * transform.up * radius + transform.position;
return position;
}
}

We can use a uniform sphere sampling for that. Given two random variables u and v (uniformly distributed), we can calculate a random point (p, q, r) on the sphere (also uniformly distributed) with:
float azimuth = v * 2.0 * PI;
float cosDistFromZenith = 1.0 - u;
float sinDistFromZenith = sqrt(1.0 - cosDistFromZenith * cosDistFromZenith);
(p, q, r) = (cos(azimuth) * sinDistFromZenith, sin(azimuth) * sinDistFromZenith, cosDistFromZenith);
If we put our reference direction (your object location) into zenithal position, we need to sample v from [0, 1] to get all directions around the object and u in [cos(minDistance), cos(maxDistance)], where minDistance and maxDistance are the angle distances from the object you want to allow. A distance of 90° or Pi/2 will give you a hemisphere. A distance of 180° or Pi will give you the full sphere.
Now that we can sample the region around the object in zenithal position, we need to account for other object locations as well. Let the object be positioned at (ox, oy, oz), which is a unit vector describing the direction from the sphere center.
We then build a local coordinate system:
rAxis = (ox, oy, oz)
pAxis = if |ox| < 0.9 : (1, 0, 0)
else : (0, 1, 0)
qAxis = normalize(cross(rAxis, pAxis))
pAxis = cross(qAxis, rAxis)
And finally, we can get our random point (x, y, z) on the sphere surface:
(x, y, z) = p * pAxis + q * qAxis + r * rAxis

Adapted from Nice Schertler, this is the code I am using
Vector3 GetRandomAroundSphere(float angleA, float angleB, Vector3 aroundPosition)
{
Assert.IsTrue(angleA >= 0 && angleB >= 0 && angleA <= 180 && angleB <= 180, "Both angles should be[0, 180]");
var v = Random.Range(0F, 1F);
var a = Mathf.Cos(Mathf.Deg2Rad * angleA);
var b = Mathf.Cos(Mathf.Deg2Rad * angleB);
float azimuth = v * 2.0F * UnityEngine.Mathf.PI;
float cosDistFromZenith = Random.Range(Mathf.Min(a, b), Mathf.Max(a, b));
float sinDistFromZenith = UnityEngine.Mathf.Sqrt(1.0F - cosDistFromZenith * cosDistFromZenith);
Vector3 pqr = new Vector3(UnityEngine.Mathf.Cos(azimuth) * sinDistFromZenith, UnityEngine.Mathf.Sin(azimuth) * sinDistFromZenith, cosDistFromZenith);
Vector3 rAxis = aroundPosition; // Vector3.up when around zenith
Vector3 pAxis = UnityEngine.Mathf.Abs(rAxis[0]) < 0.9 ? new Vector3(1F, 0F, 0F) : new Vector3(0F, 1F, 0F);
Vector3 qAxis = Vector3.Normalize(Vector3.Cross(rAxis, pAxis));
pAxis = Vector3.Cross(qAxis, rAxis);
Vector3 position = pqr[0] * pAxis + pqr[1] * qAxis + pqr[2] * rAxis;
return position;
}

Rotating cubes to face the origin using Quaternions

I'm in the process of setting up a relatively simple voxel-based world for a game. The high level idea is to first generate voxel locations following a fibonacci grid, then rotate the cubes such that the outer surface of the fibonacci grid resembles a sphere, and finally size the cubes such that they roughly cover the surface of the sphere (overlap is fine). See below the code for generating the voxels along the fibonacci grid:
public static Voxel[] CreateInitialVoxels(int numberOfPoints, int radius)
{
float goldenRatio = (1 + Mathf.Sqrt(5)) / 2;
Voxel[] voxels = new Voxel[numberOfPoints];
for (int i = 0; i < numberOfPoints; i++)
{
float n = i - numberOfPoints / 2; // Center at zero
float theta = 2 * Mathf.PI * n / goldenRatio;
float phi = (Mathf.PI / 2) + Mathf.Asin(2 * n / numberOfPoints);
voxels[i] = new Voxel(new Location(theta, phi, radius));
}
return voxels;
}
This generates a sphere that looks roughly like a staircase
So, my current approach to get this looking a bit more spherical is to basically rotate each cube in each pair of axes, then combine all of the rotations:
private void DrawVoxel(Voxel voxel, GameObject voxelContainer)
{
GameObject voxelObject = Instantiate<GameObject>(GetVoxelPrefab());
voxelObject.transform.position = voxel.location.cartesianCoordinates;
voxelObject.transform.parent = voxelContainer.transform;
Vector3 norm = voxel.location.cartesianCoordinates.normalized;
float xyRotationDegree = Mathf.Atan(norm.y / norm.x) * (180 / Mathf.PI);
float zxRotationDegree = Mathf.Atan(norm.z / norm.x) * (180 / Mathf.PI);
float yzRotationDegree = Mathf.Atan(norm.z / norm.y) * (180 / Mathf.PI);
Quaternion xyRotation = Quaternion.AngleAxis(xyRotationDegree, new Vector3(0, 0, 1));
Quaternion zxRotation = Quaternion.AngleAxis(zxRotationDegree, new Vector3(0, 1, 0));
Quaternion yzRotation = Quaternion.AngleAxis(yzRotationDegree, new Vector3(1, 0, 0));
voxelObject.transform.rotation = zxRotation * yzRotation * xyRotation;
}
The primary thing that I am getting caught on is that each of these rotations seems to work fine for me in isolation, but when combining them things tend to go a bit haywire (pictures below) I'm not sure exactly what the issue is. My best guess is that I've made some sign/rotation mismatch in my rotations so they don't combine right. I can get two working, but never all three together.
Above are the pictures of one and two successful rotations, followed by the error mode when I attempt to combine them. Any help either on telling me that the approach I'm following is too convoluted, or helping me understand what the right way to combine these rotations would be would be very helpful. Cartesian coordinate conversion below for reference.
[System.Serializable]
public struct Location
{
public float theta, phi, r;
public Vector3 polarCoordinates;
public float x, y, z;
public Vector3 cartesianCoordinates;
public Location(float theta, float phi, float r)
{
this.theta = theta;
this.phi = phi;
this.r= r;
this.polarCoordinates = new Vector3(theta, phi, r);
this.x = r * Mathf.Sin(phi) * Mathf.Cos(theta);
this.y = r * Mathf.Sin(phi) * Mathf.Sin(theta);
this.z = r * Mathf.Cos(phi);
this.cartesianCoordinates = new Vector3(x, y, z);
}
}

I managed to find a solution to this problem, though it's still not clear to me what the issue with the above code is.
Unity has an extremely handy function called Quaternion.FromToRotation that will generate the appropriate rotation if you simply pass in the appropriate destination vector.
In my case I was able to just do:
voxelObject.transform.rotation = Quaternion.FromToRotation(new Vector3(0, 0, 1), voxel.location.cartesianCoordinates);

C# finding angle between 2 given points

In the program that I'm working on, I have an object (the player) in the shape of a triangle, and that triangle is supposed to rotate always facing the mouse. given this two points I have tried different equations I've found online but non of them seem to work or at least preform well enough.
delta_x = cursor.X - pos.X;
delta_y = cursor.Y - pos.Y;
cursorAngle = (float)Math.Atan2(delta_y, delta_x) * (float)(180 / Math.PI);
this is the most efficient formula I found but it is still not working well enough, since it only faces the mouse at specific angles or distances. Cursor.X and .Y are the coordinates of the mouse and pos.X and .Y are the coordinates of the player.

I created this WinForm example that calculates the angle and distance of the mouse from the center of the form every time you move the mouse on the form. The result I display in a label.
The red dot in the center of the form is just a reference panel and has no relevance in the code.
private void f_main_MouseMove(object sender, MouseEventArgs e)
{
Point center = new Point(378, 171);
Point mouse = this.PointToClient(Cursor.Position);
lb_mouseposition.Text = $"Mouse Angle: {CalculeAngle(center, mouse)} / Distance: {CalculeDistance(center, mouse)}";
}
private double CalculeAngle(Point start, Point arrival)
{
var deltaX = Math.Pow((arrival.X - start.X), 2);
var deltaY = Math.Pow((arrival.Y - start.Y), 2);
var radian = Math.Atan2((arrival.Y - start.Y), (arrival.X - start.X));
var angle = (radian * (180 / Math.PI) + 360) % 360;
return angle;
}
private double CalculeDistance(Point start, Point arrival)
{
var deltaX = Math.Pow((arrival.X - start.X), 2);
var deltaY = Math.Pow((arrival.Y - start.Y), 2);
var distance = Math.Sqrt(deltaY + deltaX);
return distance;
}
The angle is here shown in degrees varying from 0 to 359.
I hope this helps in calculating the angle between your two points.

OpenTK GL.Translate 2D camera on GLControl

I am doing some ascii game, via using OpenTK & Winforms. And stack with Camera movement and view in 2D.
Those code i am using for mouse event and translate positions:
public static Point convertScreenToWorldCoords(int x, int y)
{
int[] viewport = new int[4];
Matrix4 modelViewMatrix, projectionMatrix;
GL.GetFloat(GetPName.ModelviewMatrix, out modelViewMatrix);
GL.GetFloat(GetPName.ProjectionMatrix, out projectionMatrix);
GL.GetInteger(GetPName.Viewport, viewport);
Vector2 mouse;
mouse.X = x;
mouse.Y = y;
Vector4 vector = UnProject(ref projectionMatrix, modelViewMatrix, new Size(viewport[2], viewport[3]), mouse);
Point coords = new Point((int)vector.X, (int)vector.Y);
return coords;
}
public static Vector4 UnProject(ref Matrix4 projection, Matrix4 view, Size viewport, Vector2 mouse)
{
Vector4 vec;
vec.X = 2.0f * mouse.X / (float)viewport.Width - 1;
vec.Y = 2.0f * mouse.Y / (float)viewport.Height - 1;
vec.Z = 0;
vec.W = 1.0f;
Matrix4 viewInv = Matrix4.Invert(view);
Matrix4 projInv = Matrix4.Invert(projection);
Vector4.Transform(ref vec, ref projInv, out vec);
Vector4.Transform(ref vec, ref viewInv, out vec);
if (vec.W > float.Epsilon || vec.W < float.Epsilon)
{
vec.X /= vec.W;
vec.Y /= vec.W;
vec.Z /= vec.W;
}
return vec;
}
//on mouse click event
Control control = sender as Control;
Point worldCoords = convertScreenToWorldCoords(e.X, control.ClientRectangle.Height - e.Y);
playerX = (int)Math.Floor((double)worldCoords.X / 9d);
playerY = (int)Math.Floor((double)worldCoords.Y / 9d);
And those code will setup my Projection, but, something wrong here...
//Set Projection
GL.MatrixMode(MatrixMode.Projection);
GL.LoadIdentity();
GL.Ortho(0, Width * charWidth * scale, Height * charHeight * scale, 0, -1, 1);
GL.MatrixMode(MatrixMode.Modelview);
GL.LoadIdentity();
GL.Translate(playerX, playerY, 0);
Well, my problem is GL.Translate. In this case, they not focused on playerX,Y and movement of "camera" seems reversed. If i am put GL.Translate(-playerX, -playerY, 0); - They seems moves correct, but anyways View seems not focused on player Object (player object should be always on center position of view, typical top-down view camera). But I dont know how to setting up them correctly. My experiments, multiple, devide, etc. with my X,Y pos - does not give me correct view. How it should be in this case?

Scaling right vector based on light direction

I'm having a quad that I constructed and I would like to scale the quad based on how much light, the problem is the dot product gives me negative values, which I can not use to scale the vectors on the other side of the quad. I have a mesh consists of 6 vertices, two quads. One of the two quads should extend or shrink based on how much is the dot product values, how would I scale one quad and shrink the other side based on that dot product value ?
float lightAngleRightVector = Vector3.Dot(lightDir.normalized, Source.transform.right.normalized);
lightAngleRightVector = Mathf.Clamp(lightAngleRightVector, 0.2f, 0.5f);
Global.Log("Light Angle Right Vecotr" + lightAngleRightVector);
// light projected left side, limit values);
if (lightAngleRightVector < 0.3f)
{
vxAbLeft = lightAngleRightVector;
vxCdRight = lightAngleRightVector - 0.1f;
}
// light projected right side
else if (lightAngleRightVector > 0.3f)
{
vxCdRight = lightAngleRightVector;
vxAbLeft = lightAngleRightVector - 0.1f;
}
Global.Log("VxCDRIGHT = " + vxCdRight);
Global.Log("vxAbLeft = " + vxAbLeft);
// add little bit shift up for fixing z-fighting
Vector3 vxPos1Top = (frontPt + new Vector3(0, mShadowOffestY, 0)) - (mRightFrontPt * vxAbLeft) * scale; // 1,2 vertices or on its left
Vector3 vxPos2Top = (mRightBackPt * vxAbLeft) * scale;
Vector3 vxPos3Top = frontPt;
Vector3 vxPos4Top = backPt;
Vector3 vxPos5Top =(mRightFrontPt * vxCdRight) * scale; // 5,6 vertices are on the right of the car
Vector3 vxPos6Top =(mRightBackPt * vxCdRight * scale);

Perhaps the scale should be abs( scale ), so it will be > 0 from the unlit side. Is that what you want?