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;
}
BackCube.position = cameraEye.position - cameraEye.forward * 2;
float back = cameraEye.position.z - 2f;
BackCube.position = new Vector3(BackCube.position.x, BackCube.position.y, back);
var lookPosBack = cameraEye.position - BackCube.position;
lookPosBack.y = 0;
var rotationBack = Quaternion.LookRotation(lookPosBack);
BackCube.rotation = Quaternion.Slerp(BackCube.rotation, rotationBack, 1);
So, I want the my BackCube to rotate towards the forward vector of cameraEye. The code above looks at the cameraEye, but not towards the forward vector of cameraEye. I want the forward vectors of both pointing at each other being 2 units apart from each other. I have control only over the BackCube
This worked!
Vector3 direction = Vector3.ProjectOnPlane(cameraEye.forward, Vector3.up);
BackCube.transform.position = cameraEye.position - direction.normalized * 2;
var lookPosBack = cameraEye.position - BackCube.position;
lookPosBack.y = 0;
var rotationBack = Quaternion.LookRotation(lookPosBack);
BackCube.rotation = Quaternion.Slerp(BackCube.rotation, rotationBack, 1);
I'm trying to generate a circular mesh made up of triangles with a common center at the center of the circle.
The mesh is generated properly, but the UVs are not and I am having some trouble understanding how to add them.
I assumed I would just copy the vertexes' pattern, but it didn't work out.
Here is the function:
private void _MakeMesh(int sides, float radius = 0.5f)
{
m_LiquidMesh.Clear();
float angleStep = 360.0f / (float) sides;
List<Vector3> vertexes = new List<Vector3>();
List<int> triangles = new List<int>();
List<Vector2> uvs = new List<Vector2>();
Quaternion rotation = Quaternion.Euler(0.0f, angleStep, 0.0f);
// Make first triangle.
vertexes.Add(new Vector3(0.0f, 0.0f, 0.0f));
vertexes.Add(new Vector3(radius, 0.0f, 0.0f));
vertexes.Add(rotation * vertexes[1]);
// First UV ??
uvs.Add(new Vector2(0, 0));
uvs.Add(new Vector2(1, 0));
uvs.Add(rotation * uvs[1]);
// Add triangle indices.
triangles.Add(0);
triangles.Add(1);
triangles.Add(2);
for (int i = 0; i < sides - 1; i++)
{
triangles.Add(0);
triangles.Add(vertexes.Count - 1);
triangles.Add(vertexes.Count);
// UV ??
vertexes.Add(rotation * vertexes[vertexes.Count - 1]);
}
m_LiquidMesh.vertices = vertexes.ToArray();
m_LiquidMesh.triangles = triangles.ToArray();
m_LiquidMesh.uv = uvs.ToArray();
m_LiquidMesh.RecalculateNormals();
m_LiquidMesh.RecalculateBounds();
Debug.Log("<color=yellow>Liquid mesh created</color>");
}
How does mapping UV work in a case like this?
Edit: I'm trying to use this circle as an effect of something flowing outwards from the center (think: liquid mesh for a brewing pot)
This is an old post, but maybe someone else will benefit from my solution.
So basically I gave my center point the center of the uv (0.5, 0.5) and then used the used circle formula to give every other point the uv coordinate. But of course I had to remap the cos and sin results from -1..1 to 0..1 and everything is working great.
Vector2[] uv = new Vector2[vertices.Length];
uv[uv.Length - 1] = new Vector2(0.5f, 0.5f);
for (int i = 0; i < uv.Length - 1; i++)
{
float radians = (float) i / (uv.Length - 1) * 2 * Mathf.PI;
uv[i] = new Vector2(Mathf.Cos(radians).Remap(-1f, 1f, 0f, 1f), Mathf.Sin(radians).Remap(-1f, 1f, 0f, 1f));
}
mesh.uv = uv;
Where the remap is an extension like this and it basically take a value in a range and remaps it to another range (in this case from -1..1 to 0..1):
public static float Remap(this float value, float from1, float to1, float from2, float to2) {
return (value - from1) / (to1 - from1) * (to2 - from2) + from2;
}
I have a simple paddle that I am trying to move along the X axis using the mouse. Currently it moves but is off by a large margin, relative to where the mouse is.
I think it is due the fact that half my screen is -7.5 and the other half is 7.5
I was wondering if there is any way to correct this problem. As you can see from my code I am multiplying by 16, which would be the width if the other half was not negative.
I can move the whole screenplay to make it not negative, so I was hoping there was a function
Vector3 paddlePos = new Vector3 (0f, this.transform.position.y , -0.25f);
float mousePosInBlocks = Input.mousePosition.x / Screen.width * 16;
paddlePos.x = Mathf.Clamp(mousePosInBlocks, -7.5f, 7.5f);
this.transform.position = paddlePos;
Use this code -
Vector3 paddlePos = new Vector3 (0f, this.transform.position.y , -0.25f);
float mousePosInBlocks = Input.mousePosition.x / Screen.width * 16;
paddlePos.x = Mathf.Clamp((mousePosInBlocks - 7.5f), -7.5f, 7.5f);
this.transform.position = paddlePos;
Mathf.clamp function only returns the value inside the minimum and maximum value range. While you need the mousePosInBlocks value to consider the negative and positive screen space.
I am having a bit of trouble figuring out how to rotate a sphere (planet) from a starting static position around it's local y-axis using a Matrix4x4 with custom rotation formulas. The planet doesn't rotate at all. I've posted the code below. Thanks for any advice.
Note: ori would be the degrees per second it would rotate
void OrientationRate()
{
Matrix4x4 o = new Matrix4x4();
float theta = ori * Time.fixedDeltaTime;
o[0, 0] = Mathf.Cos(theta);
o[0, 2] = -Mathf.Sin(theta);
o[2, 0] = Mathf.Sin(theta);
o[2, 2] = Mathf.Cos(theta);
Vector4 p = this.transform.eulerAngles;
p.w = 1;
Vector4 pprime = (o * p);
//set the new position
this.transform.Rotate(new Vector3(pprime.z, pprime.x, pprime.y) * Time.fixedDeltaTime);
}