I am creating a small demo game where you can rotate around earth and launch satellites into space. But I have some trouble with the calculations.
You can drag the mouse from the platform to a direction. This is the direction you shoot the satellite to. Because the camera is rotated around the planet, up isn't the same as forward. For the direction of the satellite, I need a Vector3 (direction/velocity).
So data I have is the forward of the platform on screen and the mosue drag direction.
So When the user drags it to (-0.7, 0.7) it means the satilatie launch direction should be (0, 0, 1). The global/World Forward direction.
So how can I translate those 2d screen position and direction to the world direction?
PlayCanvas has a very useful function we could make use of. The implementation is as follows:
* #description Convert a point from 2D canvas pixel space to 3D world space.
* #param {Number} x x coordinate on PlayCanvas' canvas element.
* #param {Number} y y coordinate on PlayCanvas' canvas element.
* #param {Number} z The distance from the camera in world space to create the new point.
* #param {Number} cw The width of PlayCanvas' canvas element.
* #param {Number} ch The height of PlayCanvas' canvas element.
* #param {pc.Vec3} [worldCoord] 3D vector to receive world coordinate result.
* #returns {pc.Vec3} The world space coordinate.
*/
screenToWorld: function (x, y, z, cw, ch, worldCoord) {
...
We can use this function to convert the start and end points (A and B respectively in the diagram) of the mouse drag line to 3D lines in world space. After the conversion we must subtract the camera's world position from the two projected points, and normalize the resulting vectors.
[The z parameter is irrelevant for this purpose because we are only interested in a direction vector and not an actual point, so just set it to e.g. 1. ]
So what does this give us? A plane spanned by these two vectors:
There are three criteria that the velocity direction must satisfy:
Perpendicular to the surface normal (i.e. tangent to the surface) at the launch site.
Parallel to the plane we just found.
Have a component in the direction from A to B.
Let:
Screen points A and B project to directional vectors U and V respectively.
The surface normal at the launch site (the "up" direction as seen by a person standing there) be N:
Where (ψ, φ) = (lat, long).
Finally, the (un-normalized) velocity direction is simply given by cross(N, cross(A, B)). Note that the order of operations matters.
To visualize this:
EDIT:
Small mistake in the second diagram: U×V should be V×U, but the expected result N×(U×V) is still correct.
Note that UxV is not necessarily perpendicular to N. When it is parallel to N, the blue plane "scrapes" the surface, i.e. the green line AB is tangent to the Earth's surface as per rendered on-screen, at the launch site.
Is there a way to convert that data:
Object position which is a 3D point (X, Y, Z),
Camera position which is a 3D point (X, Y, Z),
Camera yaw, pitch, roll (-180:180, -90:90, 0)
Field of view (-45°:45°)
Screen width & height
into the 2D point on the screen (X, Y)?
I'm looking for proper math calculations according to this exact set of data.
It's difficult, but it's possible to do it for yourself.
There are lots of libraries that do this for you, but it is more satisfying if you do it yourself:
This problem is possible and I have written my own 3D engine to do this for objects in javascript using the HTML5 Canvas. You can see my code here and solve a 3D maze game I wrote here to try and understand what I will talk about below...
The basic idea is to work in steps. To start, you have to forget about camera angle (yaw, pitch and roll) as these come later and just imagine you are looking down the y axis. Then the basic idea is to calculate, using trig, the pitch angle and yaw to your object coordinate. By this I mean imagining that you are looking through a letterbox, the yaw angle would be the angle in degrees left and right to your coordinate (so both positive and negative) from the center/ mid line and the yaw up and down from it. Taking these angles, you can map them to the x and y 2D coordinate system.
The calculations for the angles are:
pitch = atan((coord.x - cam.x) / (coord.y - cam.y))
yaw = atan((coord.z - cam.z) / (coord.y - cam.y))
with coord.x, coord.y and coord.z being the coordinates of the object and the same for the cam (cam.x, cam.y and cam.z). These calculations also assume that you are using a Cartesian coordinate system with the different axis being: z up, y forward and x right.
From here, the next step is to map this angle in the 3D world to a coordinate which you can use in a 2D graphical representation.
To map these angles into your screen, you need to scale them up as distances from the mid line. This means multiplying them by your screen width / fov. Finally, these distances will now be positive or negative (as it is an angle from the mid line) so to actually draw it on a canvas, you need to add it to half of the screen width.
So this would mean your canvas coordinate would be:
x = width / 2 + (pitch * (width / fov)
y = height / 2 + (yaw * (height / fov)
where width and height are the dimensions of you screen, fov is the camera's fov and yaw and pitch are the respective angles of the object from the camera.
You have now achieved the first big step which is mapping a 3D coordinate down to 2D. If you have managed to get this all working, I would suggest trying multiple points and connecting them to form shapes. Also try moving your cameras position to see how the perspective changes as you will soon see how realistic it already looks.
In addition, if this worked fine for you, you can move on to having the camera be able to not only change its position in the 3D world but also change its perspective as in yaw, pitch and roll angles. I will not go into this entirely now, but the basic idea is to use 3D world transformation matrices. You can read up about them here but they do get quite complicated, however I can give you the calculations if you get this far.
It might help to read (old style) OpenGL specs:
https://www.khronos.org/registry/OpenGL/specs/gl/glspec14.pdf
See section 2.10
Also:
https://www.khronos.org/opengl/wiki/Vertex_Transformation
Might help with more concrete examples.
Also, for "proper math" look up 4x4 matrices, projections, and homogeneous coordinates.
https://en.wikipedia.org/wiki/Homogeneous_coordinates
Sorry if this question has been asked before, but if so I could not find it before posting this.
In a nutshell, I want to do this:
Example.
I want a pointer (red) to rotate about the circle (blue) according to where the mouse is located. (If picture is not visible, it depicts a blue circle with a red triangle pointing away from it, towards the mouse).
If possible, please answer with a general mathematical equation rather than specific code. Thanks.
Assuming a normal cartesian coordinate space, with the X axis going to the right and the Y axis going up, you first need to calculate the angle to the mouse coordinate (M) to the circle origin (O):
theta = atan2(M.y - O.y, M.x - O.x)
you can then calculate the position of a point (P) orbiting the circle at radius (r) with:
P.x = r * cos(theta)
P.y = r * sin(theta)
The atan2(y, x) function is a common math library function that just computes atan(y / x) but takes the relative signs of x and y into account to determine the correct quadrant.
So I have a bit of a math problem. Here are the pieces.
Input:
Rot = Rotation (degrees). This is the rotation of the "player". This is also the yaw.
Vel.X = This is the left/rightward movement that would be happening if it weren't rotated
Vel.Z = Same as last except its up/down movement
Output:
Result.X = This is the actual movement that should be happening along the x axis considering rotation
Result.Z = Same as last
Basically the scenario is that a player is standing on a platform with "Rot" rotation. When directional keys are pressed velocity is added accordingly to the "Vel" value. However if rotation isn't 0 this wont produce the right result because when the player rotates moving left becomes relative.
Could you please tell me a formula that would find the proper x and y movement that would result in the player moving around relative to its rotation?
This problem is probably the most basic rotation question in game programming.
Using your Vel.X and Vel.Z values, you have what you might think of as the vector you wish to rotate in the x/z plane (instead of x/y - but same idea). Whether velocity or position, the approach is the same. With a simple google search we find that for 2D vector rotation, the formula is:
Result.X = Vel.X * cos(Rot) - Vel.Z * sin(Rot);
Result.Z = Vel.X * sin(Rot) + Vel.Z * cos(Rot);
I am writing a shader to render spheres on point sprites, by drawing shaded circles, and need to write a depth component as well as colour in order that spheres near each other will intersect correctly.
I am using code similar to that written by Johna Holwerda:
void PS_ShowDepth(VS_OUTPUT input, out float4 color: COLOR0,out float depth : DEPTH)
{
float dist = length (input.uv - float2 (0.5f, 0.5f)); //get the distance form the center of the point-sprite
float alpha = saturate(sign (0.5f - dist));
sphereDepth = cos (dist * 3.14159) * sphereThickness * particleSize; //calculate how thick the sphere should be; sphereThickness is a variable.
depth = saturate (sphereDepth + input.color.w); //input.color.w represents the depth value of the pixel on the point-sprite
color = float4 (depth.xxx ,alpha ); //or anything else you might need in future passes
}
The video at that link gives a good idea of the effect I'm after: those spheres drawn on point sprites intersect correctly. I've added images below to illustrate too.
I can calculate the depth of the point sprite itself fine. However, I am not sure show to calculate the thickness of the sphere at a pixel in order to add it to the sprite's depth, to give a final depth value. (The above code uses a variable rather than calculating it.)
I've been working on this on and off for several weeks but haven't figured it out - I'm sure it's simple, but it's something my brain hasn't twigged.
Direct3D 9's point sprite sizes are calculated in pixels, and my sprites have several sizes - both by falloff due to distance (I implemented the same algorithm the old fixed-function pipeline used for point size computations in my vertex shader) and also due to what the sprite represents.
How do I go from the data I have in a pixel shader (sprite location, sprite depth, original world-space radius, radius in pixels onscreen, normalised distance of the pixel in question from the centre of the sprite) to a depth value? A partial solution simply of sprite size to sphere thickness in depth coordinates would be fine - that can be scaled by the normalised distance from the centre to get the thickness of the sphere at a pixel.
I am using Direct3D 9 and HLSL with shader model 3 as the upper SM limit.
In pictures
To demonstrate the technique, and the point at which I'm having trouble:
Start with two point sprites, and in the pixel shader draw a circle on each, using clip to remove fragments outside the circle's boundary:
One will render above the other, since after all they are flat surfaces.
Now, make the shader more advanced, and draw the circle as though it was a sphere, with lighting. Note that even though the flat sprites look 3D, they still draw with one fully in front of the other since it's an illusion: they are still flat.
(The above is easy; it's the final step I am having trouble with and am asking how to achieve.)
Now, instead of the pixel shader writing only colour values, it should write the depth as well:
void SpherePS (...any parameters...
out float4 oBackBuffer : COLOR0,
out float oDepth : DEPTH0 <- now also writing depth
)
{
Note that now the spheres intersect when the distance between them is smaller than their radiuses:
How do I calculate the correct depth value in order to achieve this final step?
Edit / Notes
Several people have commented that a real sphere will distort due to perspective, which may be especially visible at the edges of the screen, and so I should use a different technique. First, thanks for pointing that out, it's not necessarily obvious and is good for future readers! Second, my aim is not to render a perspective-correct sphere, but to render millions of data points fast, and visually I think a sphere-like object looks nicer than a flat sprite, and shows the spatial position better too. Slight distortion or lack of distortion does not matter. If you watch the demo video, you can see how it is a useful visual tool. I don't want to render actual sphere meshes because of the large number of triangles compared to a simple hardware-generated point sprite. I really do want to use the technique of point sprites, and I simply want to extend the extant demo technique in order to calculate the correct depth value, which in the demo was passed in as a variable with no source for how it was derived.
I came up with a solution yesterday, which which works well and and produces the desired result of a sphere drawn on the sprite, with a correct depth value which intersects with other objects and spheres in the scene. It may be less efficient than it needs to be (it calculates and projects two vertices per sprite, for example) and is probably not fully correct mathematically (it takes shortcuts), but it produces visually good results.
The technique
In order to write out the depth of the 'sphere', you need to calculate the radius of the sphere in depth coordinates - i.e., how thick half the sphere is. This amount can then be scaled as you write out each pixel on the sphere by how far from the centre of the sphere you are.
To calculate the radius in depth coordinates:
Vertex shader: in unprojected scene coordinates cast a ray from the eye through the sphere centre (that is, the vertex that represents the point sprite) and add the radius of the sphere. This gives you a point lying on the surface of the sphere. Project both the sprite vertex and your new sphere surface vertex, and calculate depth (z/w) for each. The different is the depth value you need.
Pixel Shader: to draw a circle you already calculate a normalised distance from the centre of the sprite, using clip to not draw pixels outside the circle. Since it's normalised (0-1), multiply this by the sphere depth (which is the depth value of the radius, i.e. the pixel at the centre of the sphere) and add to the depth of the flat sprite itself. This gives a depth thickest at the sphere centre to 0 and the edge, following the surface of the sphere. (Depending on how accurate you need it, use a cosine to get a curved thickness. I found linear gave perfectly fine-looking results.)
Code
This is not full code since my effects are for my company, but the code here is rewritten from my actual effect file omitting unnecessary / proprietary stuff, and should be complete enough to demonstrate the technique.
Vertex shader
void SphereVS(float4 vPos // Input vertex,
float fPointRadius, // Radius of circle / sphere in world coords
out float fDXScale, // Result of DirectX algorithm to scale the sprite size
out float fDepth, // Flat sprite depth
out float4 oPos : POSITION0, // Projected sprite position
out float fDiameter : PSIZE, // Sprite size in pixels (DX point sprites are sized in px)
out float fSphereRadiusDepth : TEXCOORDn // Radius of the sphere in depth coords
{
...
// Normal projection
oPos = mul(vPos, g_mWorldViewProj);
// DX depth (of the flat billboarded point sprite)
fDepth = oPos.z / oPos.w;
// Also scale the sprite size - DX specifies a point sprite's size in pixels.
// One (old) algorithm is in http://msdn.microsoft.com/en-us/library/windows/desktop/bb147281(v=vs.85).aspx
fDXScale = ...;
fDiameter = fDXScale * fPointRadius;
// Finally, the key: what's the depth coord to use for the thickness of the sphere?
fSphereRadiusDepth = CalculateSphereDepth(vPos, fPointRadius, fDepth, fDXScale);
...
}
All standard stuff, but I include it to show how it's used.
The key method and the answer to the question is:
float CalculateSphereDepth(float4 vPos, float fPointRadius, float fSphereCenterDepth, float fDXScale) {
// Calculate sphere depth. Do this by calculating a point on the
// far side of the sphere, ie cast a ray from the eye, through the
// point sprite vertex (the sphere center) and extend it by the radius
// of the sphere
// The difference in depths between the sphere center and the sphere
// edge is then used to write out sphere 'depth' on the sprite.
float4 vRayDir = vPos - g_vecEyePos;
float fLength = length(vRayDir);
vRayDir = normalize(vRayDir);
fLength = fLength + vPointRadius; // Distance from eye through sphere center to edge of sphere
float4 oSphereEdgePos = g_vecEyePos + (fLength * vRayDir); // Point on the edge of the sphere
oSphereEdgePos.w = 1.0;
oSphereEdgePos = mul(oSphereEdgePos, g_mWorldViewProj); // Project it
// DX depth calculation of the projected sphere-edge point
const float fSphereEdgeDepth = oSphereEdgePos.z / oSphereEdgePos.w;
float fSphereRadiusDepth = fSphereCenterDepth - fSphereEdgeDepth; // Difference between center and edge of sphere
fSphereRadiusDepth *= fDXScale; // Account for sphere scaling
return fSphereRadiusDepth;
}
Pixel shader
void SpherePS(
...
float fSpriteDepth : TEXCOORD0,
float fSphereRadiusDepth : TEXCOORD1,
out float4 oFragment : COLOR0,
out float fSphereDepth : DEPTH0
)
{
float fCircleDist = ...; // See example code in the question
// 0-1 value from the center of the sprite, use clip to form the sprite into a circle
clip(fCircleDist);
fSphereDepth = fSpriteDepth + (fCircleDist * fSphereRadiusDepth);
// And calculate a pixel color
oFragment = ...; // Add lighting etc here
}
This code omits lighting etc. To calculate how far the pixel is from the centre of the sprite (to get fCircleDist) see the example code in the question (calculates 'float dist = ...') which already drew a circle.
The end result is...
Result
Voila, point sprites drawing spheres.
Notes
The scaling algorithm for the sprites may require the depth to be
scaled, too. I am not sure that line is correct.
It is not fully mathematically correct (takes shortcuts)
but as you can see the result is visually correct
When using millions of sprites, I still get a good rendering speed (<10ms per frame for 3 million sprites, on a VMWare Fusion emulated Direct3D device)
The first big mistake is that a real 3d sphere will not project to a circle under perspective 3d projection.
This is very non intuitive, but look at some pictures, especially with a large field of view and off center spheres.
Second, I would recommend against using point sprites in the beginning, it might make things harder than necessary, especially considering the first point. Just draw a generous bounding quad around your sphere and go from there.
In your shader you should have the screen space position as an input. From that, the view transform, and your projection matrix you can get to a line in eye space. You need to intersect this line with the sphere in eye space (raytracing), get the eye space intersection point, and transform that back to screen space. Then output 1/w as depth. I am not doing the math for you here because I am a bit drunk and lazy and I don't think that's what you really want to do anyway. It's a great exercise in linear algebra though, so maybe you should try it. :)
The effect you are probably trying to do is called Depth Sprites and is usually used only with an orthographic projection and with the depth of a sprite stored in a texture. Just store the depth along with your color for example in the alpha channel and just output
eye.z+(storeddepth-.5)*depthofsprite.
Sphere will not project into a circle in general case. Here is the solution.
This technique is called spherical billboards. An in-depth description can be found in this paper:
Spherical Billboards and their Application to Rendering Explosions
You draw point sprites as quads and then sample a depth texture in order to find the distance between per-pixel Z-value and your current Z-coordinate. The distance between the sampled Z-value and current Z affects the opacity of the pixel to make it look like a sphere while intersecting underlying geometry. Authors of the paper suggest the following code to compute opacity:
float Opacity(float3 P, float3 Q, float r, float2 scr)
{
float alpha = 0;
float d = length(P.xy - Q.xy);
if(d < r) {
float w = sqrt(r*r - d*d);
float F = P.z - w;
float B = P.z + w;
float Zs = tex2D(Depth, scr);
float ds = min(Zs, B) - max(f, F);
alpha = 1 - exp(-tau * (1-d/r) * ds);
}
return alpha;
}
This will prevent sharp intersections of your billboards with the scene geometry.
In case point-sprites pipeline is difficult to control (i can say only about OpenGL and not DirectX) it is better to use GPU-accelerated billboarding: you supply 4 equal 3D vertices that match the center of the particle. Then you move them into the appropriate billboard corners in a vertex shader, i.e:
if ( idx == 0 ) ParticlePos += (-X - Y);
if ( idx == 1 ) ParticlePos += (+X - Y);
if ( idx == 2 ) ParticlePos += (+X + Y);
if ( idx == 3 ) ParticlePos += (-X + Y);
This is more oriented to the modern GPU pipeline and of coarse will work with any nondegenerate perspective projection.