what I am trying to do is implementing soft shadows in my simple ray tracer, developed in C++. The idea behind this, if I understood correctly, is to shoot multiple rays towards the light, instead of a single ray towards the center of the light, and average the results. The rays are therefore shot in different positions of the light. So far I am using random points, which I don't know if it is correct or if I should use points regularly distributed on the light surface. Assuming that I am doing right, I choose a random point on the light, which in my framework is implemented as a sphere. This is given by:
Vec3<T> randomPoint() const
{
T x;
T y;
T z;
// random vector in unit sphere
std::random_device rd; //used for the new <random> library
std::mt19937 gen(rd());
std::uniform_real_distribution<> dis(-1, 1);
do
{
x = dis(gen);
y = dis(gen);
z = dis(gen);
} while (pow(x, 2) + pow(y, 2) + pow(z, 2) > 1); // simple rejection sampling
return center + Vec3<T>(x, y, z) * radius;
}
After this, I don't know how exactly I should move since my rendering equation (in my simple ray tracer) is defined as follows:
Vec3<float> surfaceColor = 0
for(int i < 0; i < lightsInTheScene.size(); i++){
surfaceColor += obj->surfaceColor * transmission *
std::max(float(0), nHit.dot(lightDirection)) * g_lights[i]->emissionColor;
}
return surfaceColor + obj->emissionColor;
where transmission is a simple float which is set to 0 in case the ray that goes from my hitPoint to the lightCenter used to find an object in the middle.
So, what I tried to do was:
creating multiple rays towards random points on the light
counting how many of them hit an object on their path and memorize this number
for simplicity: Let's imagine in my case that I shoot 3 shadow rays from my point towards random points on the light. Only 2 of 3 rays reach the light. Therefore the final color of my pixel will be = color * shadowFactor where shadowFactor = 2/3. In my equation then I delete the transmission factor (which is now wrong) and I use the shadowFactor instead. The problem is that in my equation I have:
std::max(float(0), nHit.dot(lightDirection))
Which I don't know how to change since I don't have anymore a lightDirection which points towards the center of the light. Can you please help me understanding what should I do it and what's wrong so far? Thanks in advance!
You should evaluate the entire BRDF for the picked light samples. Then, you will also have the light direction (vector from object position to picked light sample). And you can average these results. Note that most area lights have a non-isotropic light emission characteristic (i.e. the amount of light emitted from a point varies by the outgoing direction).
Averaging the visibility does not produce correct results (although they are usually visually plausible).
Related
The situation si as follows. I am trying to implement a linear voxel search in a glsl shader for efficient voxel ray tracing. In toehr words, I have a 3D texture and I am ray tracing on it but I am trying to ray trace such that I only ever check voxels intersected by the ray once.
To this effect I have written a program with the following results:
Not efficient but correct:
The above image was obtained by adding a small epsilon ray multiple times and sampling from the texture on each iteration. Which produces the correct results but it's very inefficient.
That would look like:
loop{
start += direction*0.01;
sample(start);
}
To make it efficient I decided to instead implement the following lookup function:
float bound(float val)
{
if(val >= 0)
return voxel_size;
return 0;
}
float planeIntersection(vec3 ray, vec3 origin, vec3 n, vec3 q)
{
n = normalize(n);
if(dot(ray,n)!=0)
return (dot(q,n)-dot(n,origin))/dot(ray,n);
return -1;
}
vec3 get_voxel(vec3 start, vec3 direction)
{
direction = normalize(direction);
vec3 discretized_pos = ivec3((start*1.f/(voxel_size))) * voxel_size;
vec3 n_x = vec3(sign(direction.x), 0,0);
vec3 n_y = vec3(0, sign(direction.y),0);
vec3 n_z = vec3(0, 0,sign(direction.z));
float bound_x, bound_y, bound_z;
bound_x = bound(direction.x);
bound_y = bound(direction.y);
bound_z = bound(direction.z);
float t_x, t_y, t_z;
t_x = planeIntersection(direction, start, n_x,
discretized_pos+vec3(bound_x,0,0));
t_y = planeIntersection(direction, start, n_y,
discretized_pos+vec3(0,bound_y,0));
t_z = planeIntersection(direction, start, n_z,
discretized_pos+vec3(0,0,bound_z));
if(t_x < 0)
t_x = 1.f/0.f;
if(t_y < 0)
t_y = 1.f/0.f;
if(t_z < 0)
t_z = 1.f/0.f;
float t = min(t_x, t_y);
t = min(t, t_z);
return start + direction*t;
}
Which produces the following result:
Notice the triangle aliasing on the left side of some surfaces.
It seems this aliasing occurs because some coordinates are not being set to their correct voxel.
For example modifying the truncation part as follows:
vec3 discretized_pos = ivec3((start*1.f/(voxel_size)) - vec3(0.1)) * voxel_size;
Creates:
So it has fixed the issue for some surfaces and caused it for others.
I wanted to know if there is a way in which I can correct this truncation so that this error does not happen.
Update:
I have narrowed down the issue a bit. Observe the following image:
The numbers represent the order in which I expect the boxes to be visited.
As you can see for some of the points the sampling of the fifth box seems to be ommitted.
The following is the sampling code:
vec4 grabVoxel(vec3 pos)
{
pos *= 1.f/base_voxel_size;
pos.x /= (width-1);
pos.y /= (depth-1);
pos.z /= (height-1);
vec4 voxelVal = texture(voxel_map, pos);
return voxelVal;
}
yep that was the +/- rounding I was talking about in my comments somewhere in your previous questions related to this. What you need to do is having step equal to grid size in one of the axises (and test 3 times once for |dx|=1 then for |dy|=1 and lastly |dz|=1).
Also you should create a debug draw 2D slice through your map to actually see where the hits for a single specific test ray occurred. Now based on direction of ray in each axis you set the rounding rules separately. Without this you are just blindly patching one case and corrupting other two ...
Now actually Look at this (I linked it to your before but you clearly did not):
Wolf and Doom ray casting techniques
especially pay attention to:
On the right It shows you how to compute the ray step (your epsilon). You simply scale the ray direction so one of the coordinate is +/-1. For simplicity start with 2D slice through your map. The red dot is ray start position. Green is ray step vector for vertical grid lines hits and red is for horizontal grid lines hits (z will be analogically the same).
Now you should add the 2D overview of your map through some height slice that is visible (like on the image on the left) add a dot or marker to each intersection detected but distinguish between x,y and z hits by color. Do this for single ray only (I use the center of view ray). Fist handle view when you look at X+ directions than X- and when done move to Y,Z ...
In my GLSL volumetric 3D back raytracer I also linked you before look at these lines:
if (dir.x<0.0) { p+=dir*(((floor(p.x*n)-_zero)*_n)-ray_pos.x)/dir.x; nnor=vec3(+1.0,0.0,0.0); }
if (dir.x>0.0) { p+=dir*((( ceil(p.x*n)+_zero)*_n)-ray_pos.x)/dir.x; nnor=vec3(-1.0,0.0,0.0); }
if (dir.y<0.0) { p+=dir*(((floor(p.y*n)-_zero)*_n)-ray_pos.y)/dir.y; nnor=vec3(0.0,+1.0,0.0); }
if (dir.y>0.0) { p+=dir*((( ceil(p.y*n)+_zero)*_n)-ray_pos.y)/dir.y; nnor=vec3(0.0,-1.0,0.0); }
if (dir.z<0.0) { p+=dir*(((floor(p.z*n)-_zero)*_n)-ray_pos.z)/dir.z; nnor=vec3(0.0,0.0,+1.0); }
if (dir.z>0.0) { p+=dir*((( ceil(p.z*n)+_zero)*_n)-ray_pos.z)/dir.z; nnor=vec3(0.0,0.0,-1.0); }
they are how I did this. As you can see I use different rounding/flooring rule for each of the 6 cases. This way you handle case without corrupting the other. The rounding rule depends on a lot of stuff like how is your coordinate system offseted to (0,0,0) and more so it might be different in your code but the if conditions should be the same. Also as you can see I am handling this by offsetting the ray start position a bit instead of having these conditions inside the ray traversal loop castray.
That macro cast ray and look for intersections with grid and on top of that actually zsorts the intersections and use the first valid one (that is what l,ll are for and no other conditions or combination of ray results are needed). So my way of deal with this is cast ray for each type of intersection (x,y,z) starting on the first intersection with the grid for the same axis. You need to take into account the starting offset so the l,ll resembles the intersection distance to real start of ray not to offseted one ...
Also a good idea is to do this on CPU side first and when 100% working port to GLSL as in GLSL is very hard to debug things like this.
I am building a ray tracer and I am able to correctly render diffuse and specular parts of my sphere. When I come to calculate shadows and reflections however I end up with a very pixelated result as shown in the below image:
I can see that the shadow is cast in the correct place and if you zoom in the reflection is also visible but again pixelated. I call this method to determine if a pixel is in shade and it is also called recursively by my reflect ray method to determine the reflected colours.
RGBColour Scene::illumination(Ray incidentRay, Shape *closestShape, RGBColour shapeColour, Ray ray)
{
RGBColour diffuseLight = _backgroundColour;
RGBColour specularLight = _backgroundColour;
double projectionNormalToSource = 0.0;
for (int i = 0; i < _lightSources.size(); i++)
{
Ray shadowRay(incidentRay.Direction(), (_lightSources.at(i).GetPosition() - incidentRay.Direction()).UnitVector());
Vector surfaceNormal = closestShape->SurfaceNormal(incidentRay);
//lambertian shading.
projectionNormalToSource = surfaceNormal.ScalarProduct(shadowRay.Direction());
if (projectionNormalToSource > 0)
{
bool isShadow = false;
std::vector<double> shadowIntersections;
Ray temp(incidentRay.Direction(), (_lightSources.at(i).GetPosition() - incidentRay.Direction()));
for (int j = 0; j < _sceneObjects.size(); j++)
{
shadowIntersections.push_back(_sceneObjects.at(j)->Intersection(temp));
}
//Test each point to see if it is in shadow.
for (int j = 0; j < shadowIntersections.size(); j++)
{
if (shadowIntersections.at(j) != -1)
{
if (shadowIntersections.at(j) > _epsilon && shadowIntersections.at(j) <= temp.Direction().Magnitude() && closestShape != _sceneObjects.at(j))
{
isShadow = true;
}
break;
}
}
if (!isShadow)
{
diffuseLight = diffuseLight + (closestShape->Colour() * projectionNormalToSource * closestShape->DiffuseCoefficient() * _lightSources.at(i).DiffuseIntensity());
specularLight = specularLight + specularReflection(_lightSources.at(i), projectionNormalToSource, closestShape, incidentRay, temp, ray);
}
}
}
return diffuseLight + specularLight;
}
As I am able to correctly render the spheres apart from these aspects I am convinced the problem must lie within this particular method so I have not posted the others. What I think is happening is that where the pixel values retain their initial colour instead of being shaded I must incorrectly be calculating very small values or the other option is that the calculated ray did not intersect, however I do not think the latter option is valid otherwise the same intersection method would return incorrect results elsewhere in the program but as the spheres render correctly (excluding the shading and reflection).
So typically what causes results like this and can you spot any obvious logic errors in my method?
Edit: I have moved my light source in front and I can now see that the shadow appears to be correctly cast for the green sphere and the blue one becomes pixelated. So I think on any subsequent shape iterations something must not be updating correctly.
Edit 2: The first issue has been fixed and the shadows are now not pixelated, the resolution was to move the break statement into the if statement directly preceding it. The issue that the reflections are still pixelated still occurs currently.
Pixelation like this could occur due to numerical instability. An example: Suppose you calculate an intersection point that lies on a curved surface. You then use that point as the origin of a ray (a shadow ray, for example). You would assume that the ray wouldn't intersect that curved surface, but in practice it sometimes can. You could check for this by discarding such self intersections, but that could cause problems if you decide to implement concave shapes. Another approach could be to move the origin of the generated ray along its direction vector by some infinitesimally small amount, so that no unwanted self-intersection occurs.
I'm writing a raytracer in C++, and I've been having some issue with refractions. I'm rendering a sphere and a ground plane, and the sphere should refract. However, it looks more like a sphere within a sphere: the "outer" sphere looks to be shaded properly, but not refracting, while the "inner" sphere looks like it's being self-shadowed. Here's a link to what it looks like: http://imgur.com/QVGkeBT.
Here's the relevant code.
//inside main raytrace function
if(refraction > 0.0f){ //the surface is refractive
//calculate refraction vector
Ray refract(intersection,
objList[bestObj]->refractedRay(
ray.dir,intersection,&cos_theta,&R0));
//recurse
refrColor = raytrace(refract);
}
else{ //no refraction
refrColor = background;
}
//refractedRay(vec3,vec3,float*,float*)
//...initialize variables, do geometric transforms
//into air out of obj
if(dot(ray,normal) < 0){
n1 = ior;
n2 = 1.0f;
*cos = dot(ray,-normal);
}
//into obj out of air
else{
n1 = 1.0f;
n2 = ior;
*cos = dot(ray,normal);
normal = -normal;
}
//check value under sqrt
float n = n1/n2;
float disc = 1-(pow(n,2)*(1-pow(*cos,2)));
if(disc < 0){ //total internal reflection
return ray - 2*-(*cos)*normal; //reflection vector
}
return (n*ray)+(((n*(*cos))-sqrt(disc))*normal);
The sphere used to look worse, then I remembered to normalize my vectors and it looks like this. Previously, it looked like only the inner sphere all throughout. Inside the main raytrace function, I do the refraction the same way as reflection, just using the refracted ray instead. I've also tried modifying the incoming point of intersection and ray with epsilon to check for self-refracting as you can get in shadowing.
Any help would be appreciated :)
I haven't checked your refraction formulae, but this looks wrong:
//into air out of obj
if(dot(ray,normal) < 0){
n1 = ior;
n2 = 1.0f;
*cos = dot(ray,-normal);
}
If the dot product of the incident ray and the normal is less than zero, and assuming the normal points outwards of the object (which it probably should) then this case corresponds to air -> inside, so your refractive indices should be swapped. As it is now you are rendering a sphere with ior 1 / ior and since that refractive index is less than 1 you are observing total internal reflection on the edges.
Here is one of my implementations which you can take a look at to see if anything is missing (it has more features but you should be able to identify the parts you are interested in and check it your computations match). To me it looks all right so I think fixing the refractive indices should do it.
The nondeterministic pattern in the center of the sphere, though, definitely looks like self-intersection. Make sure that in the case of reflection, you push the reflected ray outside the intersected surface slightly, and in the case of refraction, push the refracted ray inside slightly, to avoid self-intersection.
This question has one major question, and one minor question. I believe I am right in either question from my research, but not both.
For my physics loop, the first thing I do is apply a gravitational force to my TotalForce for a rigid body object. I then check for collisions using my TotalForce and my Velocity. My TotalForce is reset to (0, 0, 0) after every physics loop, although I will keep my velocity.
I am familiar with doing a collision check between a moving sphere and a static plane when using only velocity. However, what if I have other forces besides velocity, such as gravity? I put the other forces into TotalForces (right now I only have gravity). To compensate for that, when I determine that the sphere is not currently overlapping the plane, I do
Vector3 forces = (sphereTotalForces + sphereVelocity);
Vector3 forcesDT = forces * fElapsedTime;
float denom = Vec3Dot(&plane->GetNormal(), &forces);
However, this can be problematic for how I thought was suppose to be resting contact. I thought resting contact was computed by
denom * dist == 0.0f
Where dist is
float dist = Vec3Dot(&plane->GetNormal(), &spherePosition) - plane->d;
(For reference, the obvious denom * dist > 0.0f meaning the sphere is moving away from the plane)
However, this can never be true. Even when there appears to be "resting contact". This is due to my forces calculation above always having at least a .y of -9.8 (my gravity). When when moving towards a plane with a normal of (0, 1, 0) will produce a y of denom of -9.8.
My question is
1) Am I calculating resting contact correctly with how I mentioned with my first two code snippets?
If so,
2) How should my "other forces" such as gravity be used? Is my use of TotalForces incorrect?
For reference, my timestep is
mAcceleration = mTotalForces / mMass;
mVelocity += mAcceleration * fElapsedTime;
Vector3 translation = (mVelocity * fElapsedTime);
EDIT
Since it appears that some suggested changes will change my collision code, here is how i detect my collision states
if(fabs(dist) <= sphereRadius)
{ // There already is a collision }
else
{
Vector3 forces = (sphereTotalForces + sphereVelocity);
float denom = Vec3Dot(&plane->GetNormal(), &forces);
// Resting contact
if(dist == 0) { }
// Sphere is moving away from plane
else if(denom * dist > 0.0f) { }
// There will eventually be a collision
else
{
float fIntersectionTime = (sphereRadius - dist) / denom;
float r;
if(dist > 0.0f)
r = sphereRadius;
else
r = -sphereRadius;
Vector3 collisionPosition = spherePosition + fIntersectionTime * sphereVelocity - r * planeNormal;
}
}
You should use if(fabs(dist) < 0.0001f) { /* collided */ } This is to acocunt for floating point accuracies. You most certainly would not get an exact 0.0f at most angles or contact.
the value of dist if negative, is in fact the actual amount you need to shift the body back onto the surface of the plane in case it goes through the plane surface. sphere.position = sphere.position - plane.Normal * fabs(dist);
Once you have moved it back to the surface, you can optionally make it bounce in the opposite direction about the plane normal; or just stay on the plane.
parallel_vec = Vec3.dot(plane.normal, -sphere.velocity);
perpendicular_vec = sphere.velocity - parallel_vec;
bounce_velocity = parallel - perpendicular_vec;
you cannot blindly do totalforce = external_force + velocity unless everything has unit mass.
EDIT:
To fully define a plane in 3D space, you plane structure should store a plane normal vector and a point on the plane. http://en.wikipedia.org/wiki/Plane_(geometry) .
Vector3 planeToSphere = sphere.point - plane.point;
float dist = Vector3.dot(plane.normal, planeToSphere) - plane.radius;
if(dist < 0)
{
// collided.
}
I suggest you study more Maths first if this is the part you do not know.
NB: Sorry, the formatting is messed up... I cannot mark it as code block.
EDIT 2:
Based on my understanding on your code, either you are naming your variables badly or as I mentioned earlier, you need to revise your maths and physics theory. This line does not do anything useful.
float denom = Vec3Dot(&plane->GetNormal(), &forces);
A at any instance of time, a force on the sphere can be in any direction at all unrelated to the direction of travel. so denom essentially calculates the amount of force in the direction of the plane surface, but tells you nothing about whether the ball will hit the plane. e.g. gravity is downwards, but a ball can have upward velocity and hit a plane above. With that, you need to Vec3Dot(plane.normal, velocity) instead.
Alternatively, Mark Phariss and Gerhard Powell had already give you the physics equation for linear kinematics, you can use those to directly calculate future positions, velocity and time of impact.
e.g. s = 0.5 * (u + v) * t; gives the displacement after future time t. compare that displacement with distance from plane and you get whether the sphere will hit the plane. So again, I suggest you read up on http://en.wikipedia.org/wiki/Linear_motion and the easy stuff first then http://en.wikipedia.org/wiki/Kinematics .
Yet another method, if you expect or assume no other forces to act on the sphere, then you do a ray / plane collision test to find the time t at which it will hit the plane, in that case, read http://en.wikipedia.org/wiki/Line-plane_intersection .
There will always be -9.8y of gravity acting on the sphere. In the case of a suspended sphere this will result in downwards acceleration (net force is non-zero). In the case of the sphere resting on the plane this will result in the plane exerting a normal force on the sphere. If the plane was perfectly horizontal with the sphere at rest this normal force would be exactly +9.8y which would perfectly cancel the force of gravity. For a sphere at rest on a non-horizontal plane the normal force is 9.8y * cos(angle) (angle is between -90 and +90 degrees).
Things get more complicated when a moving sphere hits a plane as the normal force will depend on the velocity and the plane/sphere material properties. Depending what your application requirements are you could either ignore this or try some things with the normal forces and see how it works.
For your specific questions:
I believe contact is more specifically just when dist == 0.0f, that is the sphere and plane are making contact. I assume your collision takes into account that the sphere may move past the plane in any physics time step.
Right now you don't appear to have any normal forces being put on the sphere from the plane when they are making contact. I would do this by checking for contact (dist == 0.0f) and if true adding the normal force to the sphere. In the simple case of a falling sphere onto a near horizontal plane (angle between -90 and +90 degrees) it would just be sphereTotalForces += Vector3D(0, 9.8 * cos(angle), 0).
Edit:
From here your equation for dist to compute the distance from the edge of sphere to the plane may not be correct depending on the details of your problem and code (which isn't given). Assuming your plane goes through the origin the correct equation is:
dist = Vec3Dot(&spherePosition, &plane->GetNormal()) - sphereRadius;
This is the same as your equation if plane->d == sphereRadius. Note that if the plane is not at the origin then use:
D3DXVECTOR3 vecTemp(spherePosition - pointOnPlane);
dist = Vec3Dot(&vecTemp, &plane->GetNormal()) - sphereRadius;
The exact solution to this problem involves some pretty serious math. If you want an approximate solution I strongly recommend developing it in stages.
1) Make sure your sim works without gravity. The ball must travel through space and have inelastic (or partially elastic) collisions with angled frictionless surfaces.
2) Introduce gravity. This will change ballistic trajectories from straight lines to parabolae, and introduce sliding, but it won't have much effect on collisions.
3) Introduce static and kinetic friction (independently). These will change the dynamics of sliding. Don't worry about friction in collisions for now.
4) Give the ball angular velocity and a moment of inertia. This is a big step. Make sure you can apply torques to it and get realistic angular accelerations. Note that realistic behavior of a spinning mass can be counter-intuitive.
5) Try sliding the ball along a level surface, under gravity. If you've done everything right, its angular velocity will gradually increase and its linear velocity gradually decrease, until it breaks into a roll. Experiment with giving the ball some initial spin ("draw", "follow" or "english").
6) Try the same, but on a sloped surface. This is a relatively small step.
If you get this far you'll have a pretty realistic sim. Don't try to skip any of the steps, you'll only give yourself headaches.
Answers to your physics problems:
f = mg + other_f; // m = mass, g = gravity (9.8)
a = f / m; // a = acceleration
v = u + at; // v = new speed, u = old speed, t = delta time
s = 0.5 * (u + v) *t;
When you have a collision, you change the both speeds to 0 (or v and u = -(u * 0.7) if you want it to bounce).
Because speed = 0, the ball is standing still.
If it is 2D or 3D, then you just change the speed in the direction of the normal of the surface to 0, and keep the parallel speed the same. That will result in the ball rolling on the surface.
You must move the ball to the surface if it cuts the surface. You can make collision distance to a small amount (for example 0.001) to make sure it stay still.
http://www.physicsforidiots.com/dynamics.html#vuat
Edit:
NeHe is an amazing source of game engine design:
Here is a page on collision detection with very good descriptions:
http://nehe.gamedev.net/tutorial/collision_detection/17005/
Edit 2: (From NeHe)
double DotProduct=direction.dot(plane._Normal); // Dot Product Between Plane Normal And Ray Direction
Dsc=(plane._Normal.dot(plane._Position-position))/DotProduct; // Find Distance To Collision Point
Tc= Dsc*T / Dst
Collision point= Start + Velocity*Tc
I suggest after that to take a look at erin cato articles (the author of Box2D) and Glenn fiedler articles as well.
Gravity is a strong acceleration and results in strong forces. It is easy to have faulty simulations because of floating imprecisions, variable timesteps and euler integration, very quickly.
The repositionning of the sphere at the plane surface in case it starts to burry itself passed the plane is mandatory, I noticed myself that it is better to do it only if velocity of the sphere is in opposition to the plane normal (this can be compared to face culling in 3D rendering: do not take into account backfaced planes).
also, most physics engine stops simulation on idle bodies, and most games never take gravity into account while moving, only when falling. They use "navigation meshes", and custom systems as long as they are sure the simulated objet is sticking to its "ground".
I don't know of a flawless physics simulator out there, there will always be an integration explosion, a missed collision (look for "sweeped collision").... it takes a lot of empirical fine-tweaking.
Also I suggest you look for "impulses" which is a method to avoid to tweak manually the velocity when encountering a collision.
Also take a look to "what every computer scientist should know about floating points"
good luck, you entered a mine field, randomly un-understandable, finger biting area of numerical computer science :)
For higher fidelity (wouldn't solve your main problem), I'd change your timestep to
mAcceleration = mTotalForces / mMass;
Vector3 translation = (mVelocity * fElapsedTime) + 0.5 * mAcceleration * pow(fElapsedTime, 2);
mVelocity += mAcceleration * fElapsedTime;
You mentioned that the sphere was a rigid body; are you also modeling the plane as rigid? If so, you'd have an infinite point force at the moment of contact & perfectly elastic collision without some explicit dissipation of momentum.
Force & velocity cannot be summed (incompatible units); if you're just trying to model the kinematics, you can disregard mass and work with acceleration & velocity only.
Assuming the sphere is simply dropped onto a horizontal plane with a perfectly inelastic collision (no bounce), you could do [N.B., I don't really know C syntax, so this'll be Pythonic]
mAcceleration = if isContacting then (0, 0, 0) else (0, -9.8, 0)
If you add some elasticity (say half momentum conserved) to the collision, it'd be more like
mAcceleration = (0, -9.8, 0) + if isContacting then (0, 4.9, 0)
I am developing a small tool for 3D visualization of molecules.
For my project i choose to make a thing in the way of what Mr "Brad Larson" did with his Apple software "Molecules". A link where you can find a small presentation of the technique used : Brad Larsson software presentation
For doing my job i must compute sphere impostor and cylinder impostor.
For the moment I have succeed to do the "Sphere Impostor" with the help of another tutorial Lies and Impostors
for summarize the computing of the sphere impostor : first we send a "sphere position" and the "sphere radius" to the "vertex shader" which will create in the camera-space an square which always face the camera, after that we send our square to the fragment shader where we use a simple ray tracing to find which fragment of the square is included in the sphere, and finally we compute the normal and the position of the fragment to compute lighting. (another thing we also write the gl_fragdepth for giving a good depth to our impostor sphere !)
But now i am blocked in the computing of the cylinder impostor, i try to do a parallel between the sphere impostor and the cylinder impostor but i don't find anything, my problem is that for the sphere it was some easy because the sphere is always the same no matter how we see it, we will always see the same thing : "a circle" and another thing is that the sphere was perfectly defined by Math then we can find easily the position and the normal for computing lighting and create our impostor.
For the cylinder it's not the same thing, and i failed to find a hint to modeling a form which can be used as "cylinder impostor", because the cylinder shows many different forms depending on the angle we see it !
so my request is to ask you about a solution or an indication for my problem of "cylinder impostor".
In addition to pygabriels answer I want to share a standalone implementation using the mentioned shader code from Blaine Bell (PyMOL, Schrödinger, Inc.).
The approach, explained by pygabriel, also can be improved. The bounding box can be aligned in such a way, that it always faces to the viewer. Only two faces are visible at most. Hence, only 6 vertices (ie. two faces made up of 4 triangles) are needed.
See picture here, the box (its direction vector) always faces to the viewer:
Image: Aligned bounding box
For source code, download: cylinder impostor source code
The code does not cover round caps and orthographic projections. It uses geometry shader for vertex generation. You can use the shader code under the PyMOL license agreement.
I know this question is more than one-year old, but I'd still like to give my 2 cents.
I was able to produce cylinder impostors with another technique, I took inspiration from pymol's code. Here's the basic strategy:
1) You want to draw a bounding box (a cuboid) for the cylinder. To do that you need 6 faces, that translates in 18 triangles that translates in 36 triangle vertices. Assuming that you don't have access to geometry shaders, you pass to a vertex shader 36 times the starting point of the cylinder, 36 times the direction of the cylinder, and for each of those vertex you pass the corresponding point of the bounding box. For example a vertex associated with point (0, 0, 0) means that it will be transformed in the lower-left-back corner of the bounding box, (1,1,1) means the diagonally opposite point etc..
2) In the vertex shader, you can construct the points of the cylinder, by displacing each vertex (you passed 36 equal vertices) according to the corresponding points you passed in.
At the end of this step you should have a bounding box for the cylinder.
3) Here you have to reconstruct the points on the visible surface of the bounding box. From the point you obtain, you have to perform a ray-cylinder intersection.
4) From the intersection point you can reconstruct the depth and the normal. You also have to discard intersection points that are found outside of the bounding box (this can happen when you view the cylinder along its axis, the intersection point will go infinitely far).
By the way it's a very hard task, if somebody is interested here's the source code:
https://github.com/chemlab/chemlab/blob/master/chemlab/graphics/renderers/shaders/cylinderimp.frag
https://github.com/chemlab/chemlab/blob/master/chemlab/graphics/renderers/shaders/cylinderimp.vert
A cylinder impostor can actually be done just the same way as a sphere, like Nicol Bolas did it in his tutorial. You can make a square facing the camera and colour it that it will look like a cylinder, just the same way as Nicol did it for spheres. And it's not that hard.
The way it is done is ray-tracing of course. Notice that a cylinder facing upwards in camera space is kinda easy to implement. For example intersection with the side can be projected to the xz plain, it's a 2D problem of a line intersecting with a circle. Getting the top and bottom isn't harder either, the z coordinate of the intersection is given, so you actually know the intersection point of the ray and the circle's plain, all you have to do is to check if its inside the circle. And basically, that's it, you get two points, and return the closer one (the normals are pretty trivial too).
And when it comes to an arbitrary axis, it turns out to be almost the same problem. When you solve equations at the fixed axis cylinder, you are solving them for a parameter that describes how long do you have to go from a given point in a given direction to reach the cylinder. From the "definition" of it, you should notice that this parameter doesn't change if you rotate the world. So you can rotate the arbitrary axis to become the y axis, solve the problem in a space where equations are easier, get the parameter for the line equation in that space, but return the result in camera space.
You can download the shaderfiles from here. Just an image of it in action:
The code where the magic happens (It's only long 'cos it's full of comments, but the code itself is max 50 lines):
void CylinderImpostor(out vec3 cameraPos, out vec3 cameraNormal)
{
// First get the camera space direction of the ray.
vec3 cameraPlanePos = vec3(mapping * max(cylRadius, cylHeight), 0.0) + cameraCylCenter;
vec3 cameraRayDirection = normalize(cameraPlanePos);
// Now transform data into Cylinder space wherethe cyl's symetry axis is up.
vec3 cylCenter = cameraToCylinder * cameraCylCenter;
vec3 rayDirection = normalize(cameraToCylinder * cameraPlanePos);
// We will have to return the one from the intersection of the ray and circles,
// and the ray and the side, that is closer to the camera. For that, we need to
// store the results of the computations.
vec3 circlePos, sidePos;
vec3 circleNormal, sideNormal;
bool circleIntersection = false, sideIntersection = false;
// First check if the ray intersects with the top or bottom circle
// Note that if the ray is parallel with the circles then we
// definitely won't get any intersection (but we would divide with 0).
if(rayDirection.y != 0.0){
// What we know here is that the distance of the point's y coord
// and the cylCenter is cylHeight, and the distance from the
// y axis is less than cylRadius. So we have to find a point
// which is on the line, and match these conditions.
// The equation for the y axis distances:
// rayDirection.y * t - cylCenter.y = +- cylHeight
// So t = (+-cylHeight + cylCenter.y) / rayDirection.y
// About selecting the one we need:
// - Both has to be positive, or no intersection is visible.
// - If both are positive, we need the smaller one.
float topT = (+cylHeight + cylCenter.y) / rayDirection.y;
float bottomT = (-cylHeight + cylCenter.y) / rayDirection.y;
if(topT > 0.0 && bottomT > 0.0){
float t = min(topT,bottomT);
// Now check for the x and z axis:
// If the intersection is inside the circle (so the distance on the xz plain of the point,
// and the center of circle is less than the radius), then its a point of the cylinder.
// But we can't yet return because we might get a point from the the cylinder side
// intersection that is closer to the camera.
vec3 intersection = rayDirection * t;
if( length(intersection.xz - cylCenter.xz) <= cylRadius ) {
// The value we will (optianally) return is in camera space.
circlePos = cameraRayDirection * t;
// This one is ugly, but i didn't have better idea.
circleNormal = length(circlePos - cameraCylCenter) <
length((circlePos - cameraCylCenter) + cylAxis) ? cylAxis : -cylAxis;
circleIntersection = true;
}
}
}
// Find the intersection of the ray and the cylinder's side
// The distance of the point and the y axis is sqrt(x^2 + z^2), which has to be equal to cylradius
// (rayDirection.x*t - cylCenter.x)^2 + (rayDirection.z*t - cylCenter.z)^2 = cylRadius^2
// So its a quadratic for t (A*t^2 + B*t + C = 0) where:
// A = rayDirection.x^2 + rayDirection.z^2 - if this is 0, we won't get any intersection
// B = -2*rayDirection.x*cylCenter.x - 2*rayDirection.z*cylCenter.z
// C = cylCenter.x^2 + cylCenter.z^2 - cylRadius^2
// It will give two results, we need the smaller one
float A = rayDirection.x*rayDirection.x + rayDirection.z*rayDirection.z;
if(A != 0.0) {
float B = -2*(rayDirection.x*cylCenter.x + rayDirection.z*cylCenter.z);
float C = cylCenter.x*cylCenter.x + cylCenter.z*cylCenter.z - cylRadius*cylRadius;
float det = (B * B) - (4 * A * C);
if(det >= 0.0){
float sqrtDet = sqrt(det);
float posT = (-B + sqrtDet)/(2*A);
float negT = (-B - sqrtDet)/(2*A);
float IntersectionT = min(posT, negT);
vec3 Intersect = rayDirection * IntersectionT;
if(abs(Intersect.y - cylCenter.y) < cylHeight){
// Again it's in camera space
sidePos = cameraRayDirection * IntersectionT;
sideNormal = normalize(sidePos - cameraCylCenter);
sideIntersection = true;
}
}
}
// Now get the results together:
if(sideIntersection && circleIntersection){
bool circle = length(circlePos) < length(sidePos);
cameraPos = circle ? circlePos : sidePos;
cameraNormal = circle ? circleNormal : sideNormal;
} else if(sideIntersection){
cameraPos = sidePos;
cameraNormal = sideNormal;
} else if(circleIntersection){
cameraPos = circlePos;
cameraNormal = circleNormal;
} else
discard;
}
From what I can understand of the paper, I would interpret it as follows.
An impostor cylinder, viewed from any angle has the following characteristics.
From the top, it is a circle. So considering you'll never need to view a cylinder top down, you don't need to render anything.
From the side, it is a rectangle. The pixel shader only needs to compute illumination as normal.
From any other angle, it is a rectangle (the same one computed in step 2) that curves. Its curvature can be modeled inside the pixel shader as the curvature of the top ellipse. This curvature can be considered as simply an offset of each "column" in texture space, depending on viewing angle. The minor axis of this ellipse can be computed by multiplying the major axis (thickness of the cylinder) with a factor of the current viewing angle (angle / 90), assuming that 0 means you're viewing the cylinder side-on.
Viewing angles. I have only taken the 0-90 case into account in the math below, but the other cases are trivially different.
Given the viewing angle (phi) and the diameter of the cylinder (a) here's how the shader needs to warp the Y-Axis in texture space Y = b' sin(phi). And b' = a * (phi / 90). The cases phi = 0 and phi = 90 should never be rendered.
Of course, I haven't taken the length of this cylinder into account - which would depend on your particular projection and is not an image-space problem.