OpenGL shadow mapping - shadow map texture doesn't get sampled at all? - c++

I'm currently working on an OpenGL project and I'm trying to get shadow mapping to work properly. I could get to a point where the shadow map gets rendered into a texture, but it doesn't seem to get applied to the scenery when rendered. Here's the most important bits of my code:
The shadow map vertex shader, basically a simple pass through shader (also does some additional stuff like normals, but that shouldn't distract you); it basically just transforms the vertices so they're seen from the perspective of the light (it's a directional light but since we need to assume a position, it's basically a point far away):
#version 430 core
layout(location = 0) in vec3 v_position;
layout(location = 1) in vec3 v_normal;
layout(location = 2) in vec3 v_texture;
layout(location = 3) in vec4 v_color;
out vec3 f_texture;
out vec3 f_normal;
out vec4 f_color;
uniform mat4 modelMatrix;
uniform mat4 depthViewMatrix;
uniform mat4 depthProjectionMatrix;
// Shadow map vertex shader.
void main() {
mat4 mvp = depthProjectionMatrix * depthViewMatrix * modelMatrix;
gl_Position = mvp * vec4(v_position, 1.0);
// Passing attributes on to the fragment shader
f_texture = v_texture;
f_normal = (transpose(inverse(modelMatrix)) * vec4(v_normal, 1.0)).xyz;
f_color = v_color;
}
The shadow map fragment shader that writes the depth value to a texture:
#version 430 core
layout(location = 0) out float fragmentDepth;
in vec3 f_texture;
in vec3 f_normal;
in vec4 f_color;
uniform vec3 lightDirection;
uniform sampler2DArray texSampler;
// Shadow map fragment shader.
void main() {
fragmentDepth = gl_FragCoord.z;
}
The vertex shader that actually renders the scene, but also calculates the position of the current vertex from the lights point of view (shadowCoord) to compare against the depth texture; it also applies a bias matrix, since the coordinates aren't in the correct [0, 1] interval for sampling:
#version 430 core
layout(location = 0) in vec3 v_position;
layout(location = 1) in vec3 v_normal;
layout(location = 2) in vec3 v_texture;
layout(location = 3) in vec4 v_color;
out vec3 f_texture;
out vec3 f_normal;
out vec4 f_color;
out vec3 f_shadowCoord;
uniform mat4 modelMatrix;
uniform mat4 viewMatrix;
uniform mat4 projectionMatrix;
uniform mat4 depthViewMatrix;
uniform mat4 depthProjectionMatrix;
// Simple vertex shader.
void main() {
mat4 mvp = projectionMatrix * viewMatrix * modelMatrix;
gl_Position = mvp * vec4(v_position, 1.0);
// This bias matrix adjusts the projection of a given vertex on a texture to be within 0 and 1 for proper sampling
mat4 depthBias = mat4(0.5, 0.0, 0.0, 0.5,
0.0, 0.5, 0.0, 0.5,
0.0, 0.0, 0.5, 0.5,
0.0, 0.0, 0.0, 1.0);
mat4 depthMVP = depthProjectionMatrix * depthViewMatrix * modelMatrix;
mat4 biasedDMVP = depthBias * depthMVP;
// Passing attributes on to the fragment shader
f_shadowCoord = (biasedDMVP * vec4(v_position, 1.0)).xyz;
f_texture = v_texture;
f_normal = (transpose(inverse(modelMatrix)) * vec4(v_normal, 1.0)).xyz;
f_color = v_color;
}
The fragment shader that applies textures from a texture array and receives the depth texture (uniform sampler2D shadowMap) and checks if a fragment is behind something:
#version 430 core
in vec3 f_texture;
in vec3 f_normal;
in vec4 f_color;
in vec3 f_shadowCoord;
out vec4 color;
uniform vec3 lightDirection;
uniform sampler2D shadowMap;
uniform sampler2DArray tileTextureArray;
// Very basic fragment shader.
void main() {
float visibility = 1.0;
if (texture(shadowMap, f_shadowCoord.xy).z < f_shadowCoord.z) {
visibility = 0.5;
}
color = texture(tileTextureArray, f_texture) * visibility;
}
And finally: the function that renders multiple chunks to generate the shadow map and then renders the scene with the shadow map applied:
// Generating the shadow map
glBindFramebuffer(GL_FRAMEBUFFER, m_framebuffer);
glActiveTexture(GL_TEXTURE0);
glBindTexture(GL_TEXTURE_2D, m_depthTexture);
m_shadowShader->bind();
glViewport(0, 0, 1024, 1024);
glDisable(GL_CULL_FACE);
glm::vec3 lightDir = glm::vec3(1.0f, -0.5f, 1.0f);
glm::vec3 sunPosition = FPSCamera::getPosition() - lightDir * 64.0f;
glm::mat4 depthViewMatrix = glm::lookAt(sunPosition, FPSCamera::getPosition(), glm::vec3(0, 1, 0));
glm::mat4 depthProjectionMatrix = glm::ortho<float>(-100.0f, 100.0f, -100.0f, 100.0f, 0.1f, 800.0f);
m_shadowShader->setUniformMatrix("depthViewMatrix", depthViewMatrix);
m_shadowShader->setUniformMatrix("depthProjectionMatrix", depthProjectionMatrix);
for (Chunk *chunk : m_chunks) {
m_shadowShader->setUniformMatrix("modelMatrix", chunk->getModelMatrix());
chunk->drawElements();
}
m_shadowShader->unbind();
glBindFramebuffer(GL_FRAMEBUFFER, 0);
// Normal draw call
m_chunkShader->bind();
glEnable(GL_CULL_FACE);
glViewport(0, 0, Window::getWidth(), Window::getHeight());
glm::mat4 viewMatrix = FPSCamera::getViewMatrix();
glm::mat4 projectionMatrix = FPSCamera::getProjectionMatrix();
glActiveTexture(GL_TEXTURE0);
glBindTexture(GL_TEXTURE_2D, m_depthTexture);
glActiveTexture(GL_TEXTURE1);
m_textures->bind();
m_chunkShader->setUniformMatrix("depthViewMatrix", depthViewMatrix);
m_chunkShader->setUniformMatrix("depthProjectionMatrix", depthProjectionMatrix);
m_chunkShader->setUniformMatrix("viewMatrix", viewMatrix);
m_chunkShader->setUniformMatrix("projectionMatrix", projectionMatrix);
m_chunkShader->setUniformVec3("lightDirection", lightDir);
m_chunkShader->setUniformInteger("shadowMap", 0);
m_chunkShader->setUniformInteger("tileTextureArray", 1);
for (Chunk *chunk : m_chunks) {
m_chunkShader->setUniformMatrix("modelMatrix", chunk->getModelMatrix());
chunk->drawElements();
}
Most of the code should be self-explanatory, I'm binding a FBO with a texture attached, we do a normal rendering call into the framebuffer, it gets rendered into a texture and then I'm trying to pass it into the shader for normal rendering. I've tested whether the texture gets properly generated and it does: See the generated shadow map here
However, when rendering the scene, all I see is this.
No shadows applied, visibility is 1.0 everywhere. I also use a debug context which works properly and logs errors when there are any, but it seems to be completely fine, no warnings or errors, so I'm the one doing something terribly wrong here. I'm on OpenGL 4.3 by the way.
Hopefully one of you can help me out on this, I've never got shadow maps to work before, this is the closest I've ever come, lol. Thanks in advance.

Commonly a mat4 OpenGL transformation matrix looks like this:
( X-axis.x, X-axis.y, X-axis.z, 0 )
( Y-axis.x, Y-axis.y, Y-axis.z, 0 )
( Z-axis.x, Z-axis.y, Z-axis.z, 0 )
( trans.x, trans.y, trans.z, 1 )
So your depthBias matrix, which you use to convert from normalized device coordinates (in ranage [-1, 1]) to texture coordinates (in range [0, 1]), should look like this:
mat4 depthBias = mat4(0.5, 0.0, 0.0, 0.0,
0.0, 0.5, 0.0, 0.0,
0.0, 0.0, 0.5, 0.0,
0.5, 0.5, 0.5, 1.0);
or this:
mat4 depthBias = mat4(
vec4( 0.5, 0.0, 0.0, 0.0 ),
vec4( 0.0, 0.5, 0.0, 0.0 ),
vec4( 0.0, 0.0, 0.5, 0.0 ),
vec4( 0.5, 0.5, 0.5, 1.0 ) );
After you have transformed a vertex position by the model matrix, the view matrix and the projection matrix, the vertex position is in clip space (homogeneous coordinates). You have to convert from clip space to normalized device coordinates (cartesian coordinates in range [-1, 1]). This can be done by dividing, by the w component of the homogeneous coordinate:
mat4 depthMVP = depthProjectionMatrix * depthViewMatrix * modelMatrix;
vec4 clipPos = depthMVP * vec4(v_position, 1.0);
vec4 ndcPos = vec4(clipPos.xyz / clipPos.w, 1.0);
f_shadowCoord = (depthBias * ndcPos).xyz;
A depth texture has one channel only. If you read data from the depth texture, then the data is contained in the x (or r) component of the vector.
Adapt the fragment shader code like this:
if ( texture(shadowMap, f_shadowCoord.xy).x < f_shadowCoord.z)
visibility = 0.5;
The Image Format specification of Khronos group says:
Image formats do not have to store each component. When the shader
samples such a texture, it will still resolve to a 4-value RGBA
vector. The components not stored by the image format are filled in
automatically. Zeros are used if R, G, or B is missing, while a
missing Alpha always resolves to 1.
see further:
Data Type (GLSL)
GLSL Programming/Vector and Matrix Operations
Transform the modelMatrix
How to render depth linearly in modern OpenGL with gl_FragCoord.z in fragment shader?
OpenGL Shadow map problems
Addition to the solution:
This is an important part of the solution, but there was another step needed to properly render the shadow map. The second mistake was using the wrong component of the texture to compare against f_shadowCoord.z: it should've been
texture(shadowMap, f_shadowCoord.xy).r
instead of
texture(shadowMap, f_shadowCoord.xy).z

Related

Color only the face red that is in front of camera?

I am pretty newbie to GLSL and I have been struggling with trying to find a way to color the face of my rotating cube red. I have achieved drawing red on one of the faces getting the normal x but my objective is to make the cube draw a red face to whatever face is facing front to the camera.
Fragment Shader
#version 330
in vec3 normal;
out vec4 fragColor;
in vec4 color;
in vec4 vertexColor;
void main() {
vec3 nor = normalize(normal);
fragColor = vec4(nor.x, 0.0, 0.0, 1.0);
}
Vertex Shader
#version 330
uniform mat4 u_m_matrix;
uniform mat4 u_vp_matrix;
layout (location=0) in vec3 a_position;
layout (location=1) in vec3 a_normal;
out vec3 normal;
out vec4 fragColor;
out vec4 vertexColor;
out vec4 color;
void main()
{
normal = a_normal;
gl_Position = u_vp_matrix * u_m_matrix * vec4(a_position, 1.0);
}
I tried messing with the dot product with normals and the direction at which the camera is looking (0, 0, 1) but I have not achieved anything yet.
This would be the desired effect:
I do in fact think it involves some mathematics ( dot product ), maybe getting the cosTetha and seeing if the vector is completdly perpendicular or not and depending on that drawing the faces red or black?
I view space, the z axis points out of the view port. When a side of the cube faces the camera, then the normal vector in view space is (0, 0, 1). The red color can be get form the z component of the normal vector.
But the normal vector has to be transformed from modle space to view space (in the vertex shader). For that you have to know the view matrix:
mat3 normalMat = inverse(transpose(mat3(u_v_matrix * u_m_matrix)));
normal = normalMat * a_normal;
In the fragment shader, the red color channel can be get from the z component:
vec3 nor = normalize(normal);
fragColor = vec4(nor.z, 0.0, 0.0, 1.0);
You can approximate a normal vector in normalized device space, by transforming with mat3(u_vp_matrix * u_m_matrix). That's inaccurate, but it tints the faces dependent on its orientation, too. In normalized device space, the z axis points into the viewport. e.g:
Vertex shader:
#version 330
uniform mat4 u_m_matrix;
uniform mat4 u_vp_matrix;
layout (location=0) in vec3 a_position;
layout (location=1) in vec3 a_normal;
out vec3 normal;
void main()
{
normal = mat3(u_vp_matrix * u_m_matrix) * a_normal;
gl_Position = u_vp_matrix * u_m_matrix * vec4(a_position, 1.0);
}
Fragment shader:
#version 330
in vec3 normal;
out vec4 fragColor;
void main() {
vec3 nor = normalize(normal);
fragColor = vec4(-nor.z, 0.0, 0.0, 1.0);
}
If you just want to color the face which that faces the camera, then you have to compare the cosine of the angle between the normal vector of the face and the view space z axis by the cosine of 45°. step compares a value to a edge and returns 0.0 or 1.0, dependent on the result:
Vertex shader:
mat3 normalMat = inverse(transpose(mat3(u_v_matrix * u_m_matrix)));
normal = normalMat * a_normal;
Fragment shader:
vec3 nor = normalize(normal);
//float red = step(0.707, abs(dot(nor, vec3(0.0, 0.0, 1.0))));
float red = step(0.707, dot(nor, vec3(0.0, 0.0, 1.0)));
fragColor = vec4(red, 0.0, 0.0, 1.0);

Scale 2D Texture to model scaling to prevent streching

I have an OpenGL 3.3 program whichts has different objects in, for example a simple cube. The cube's dimensions are 1x1x1 (vertices from -0.5, -0.5, -0.5 to 0.5, 0.5, 0.5) and is textured with one 2D texture on each side. The texture is repeatable (seamless).
With my actual code the model scaling looks like this (ignore the actual texture):
After scaling like this:
In this case the texture in should stay at size in z-direction but repeate over the z-axis.
Is there a good way to scale the texture properly to the model's scaling to prevent it from stretching? Or do I have to create a 3D texture?
The problem i found is that in my shader I get only the (scaled) point of the cube, for example -0.5, -1,5, -0.5 but the texture's coordinates are only 2D (0.0, 0.0) and I don't know which side of the texture I have to scale since I don't know which side it will currently be rendered on.
For for the sake of completeness, however, the vertex shader code:
layout (location = 0) in vec3 aPos;
layout (location = 1) in vec3 aNormal;
layout (location = 2) in vec2 aTexCoord;
out vec2 TexCoord;
out vec3 FragPos;
out vec3 Normal;
uniform mat4 model;
uniform mat4 view;
uniform mat4 projection;
void main()
{
FragPos = vec3(model * vec4(aPos, 1.0));
Normal = mat3(transpose(inverse(model))) * aNormal;
TexCoord = aTexCoord;
gl_Position = projection * view * model * vec4(aPos, 1.0);
//gl_Position = projection * view * model * vec4(aPos, 1.0f);
//TexCoord = aTexCoord;
}
The fragment shader looks like this:
out vec4 FragColor;
in vec2 TexCoord;
// texture samplers
uniform sampler2D texture_diffuse1;
uniform vec4 color;
void main()
{
FragColor = color + texture(texture_diffuse1, TexCoord);
}

Diffuse lighting error on parallel surfaces

As a test, I created a simple quad. Here are its attributes:
Vertex vertices[] =
{
// Positions Normals
{vec3(-1,-1, 0), vec3(-1,-1, 1)}, // v0
{vec3( 1,-1, 0), vec3( 1,-1, 1)}, // v1
{vec3(-1, 1, 0), vec3(-1, 1, 1)}, // v2
{vec3( 1, 1, 0), vec3( 1, 1, 1)}, // v3
};
And I put it in my world space at (0.0, 0.0, -9.5). Then I put my point light position at (0.0, 0.0, -8.0). My camera is at the origin (0.0, 0.0, 0.0). When I run my program, this works as expected:
But then, when I replace this quad with 9 scaled down quads, put them all at -9.5 on Z (in other word, they are all parallel to each other on Z), my diffuse lighting gets a little weird
It looks like the corners are showing too much lighting, breaking the nice diffuse circle that we see on a regular quad.
Here is my shader program:
precision mediump int;
precision mediump float;
varying vec3 v_position;
varying vec3 v_normal;
#if defined(VERTEX)
uniform mat4 u_mvpMatrix;
uniform mat4 u_mvMatrix;
uniform mat3 u_normalMatrix;
attribute vec4 a_position;
attribute vec3 a_normal;
void main()
{
vec4 position = u_mvMatrix * a_position;
v_position = position.xyz / position.w;
v_normal = normalize(u_normalMatrix * a_normal);
gl_Position = u_mvpMatrix * a_position;
}
#endif // VERTEX
#if defined(FRAGMENT)
uniform vec3 u_pointLightPosition;
void main()"
{
vec3 viewDir = normalize(-v_position);
vec3 normal = normalize(v_normal);
vec3 lightPosition = u_pointLightPosition - v_position;
vec3 pointLightDir = normalize(lightPosition);
float distance = length(lightPosition);
float pointLightAttenuation = 1.0 / (1.0 + (0.25 * distance * distance));
float diffuseTerm = max(dot(pointLightDir, normal), 0.15);
gl_FragColor = vec4(diffuseTerm * pointLightAttenuation);
}
#endif // FRAGMENT
My uniforms are uploaded as followed (I'm using GLM):
const mat4 &view_matrix = getViewMatrix();
mat4 mv_matrix = view * getModelMatrix();
mat4 mvp_matrix = getProjectionMatrix() * mv_matrix;
mat3 normal_matrix = inverseTranspose(mat3(mv_matrix));
vec3 pointLightPos = vec3(view_matrix * vec4(getPointLightPos(), 1.0f));
glUniformMatrix4fv( mvpMatrixUniformID, 1, GL_FALSE, (GLfloat*)&mvp_matrix);
glUniformMatrix4fv( vpMatrixUniformID, 1, GL_FALSE, (GLfloat*)&mv_matrix);
glUniformMatrix3fv(normalMatrixUniformID, 1, GL_FALSE, (GLfloat*)&normal_matrix);
glUniform3f(pointLightPosUniformID, pointLightPos.x, pointLightPos.y, pointLightPos.z);
Am I doing anything wrong?
Thanks!
Without going too much into your code, I think everything is working just fine. I see a very similar result with a quick blender setup:
The issue is the interpolation of the normal doesn't produce a spherical bump.
It ends up being a patch like this (I simply subdivided a smooth shaded cube)...
If you want a more spherical bump, you could generate the normals implicitly in a fragment shader (for example as is done here (bottom image)), use a normal map, or use more tessellated geometry such as an actual sphere.

Lighting doesn't show in OpenGL

I'm trying to do point source directional lighting in OpenGL using my textbooks examples. I'm showing a rectangle centered at the origin, and doing the lighting computations in the shader. The rectangle appears, but it is black even when I try to put colored lights on it. Normals for the rectangle are all (0, 1.0, 0). I'm not doing any non-uniform scaling, so the regular model view matrix should also transform the normals.
I have code that sets the light parameters(as uniforms) and material parameters(also as uniforms) for the shader. There is no per vertex color information.
void InitMaterial()
{
color material_ambient = color(1.0, 0.0, 1.0);
color material_diffuse = color(1.0, 0.8, 0.0);
color material_specular = color(1.0, 0.8, 0.0);
float material_shininess = 100.0;
// set uniforms for current program
glUniform3fv(glGetUniformLocation(Programs[lightingType], "materialAmbient"), 1, material_ambient);
glUniform3fv(glGetUniformLocation(Programs[lightingType], "materialDiffuse"), 1, material_diffuse);
glUniform3fv(glGetUniformLocation(Programs[lightingType], "materialSpecular"), 1, material_specular);
glUniform1f(glGetUniformLocation(Programs[lightingType], "shininess"), material_shininess);
}
For the lights:
void InitLight()
{
// need light direction and light position
point4 light_position = point4(0.0, 0.0, -1.0, 0.0);
color light_ambient = color(0.2, 0.2, 0.2);
color light_diffuse = color(1.0, 1.0, 1.0);
color light_specular = color(1.0, 1.0, 1.0);
glUniform3fv(glGetUniformLocation(Programs[lightingType], "lightPosition"), 1, light_position);
glUniform3fv(glGetUniformLocation(Programs[lightingType], "lightAmbient"), 1, light_ambient);
glUniform3fv(glGetUniformLocation(Programs[lightingType], "lightDiffuse"), 1, light_diffuse);
glUniform3fv(glGetUniformLocation(Programs[lightingType], "lightSpecular"), 1, light_specular);
}
The fragment shader is a simple pass through shader that sets the color to the one input from the vertex shader. Here is the vertex shader :
#version 150
in vec4 vPosition;
in vec3 vNormal;
out vec4 color;
uniform vec4 materialAmbient, materialDiffuse, materialSpecular;
uniform vec4 lightAmbient, lightDiffuse, lightSpecular;
uniform float shininess;
uniform mat4 modelView;
uniform vec4 lightPosition;
uniform mat4 projection;
void main()
{
// Transform vertex position into eye coordinates
vec3 pos = (modelView * vPosition).xyz;
vec3 L = normalize(lightPosition.xyz - pos);
vec3 E = normalize(-pos);
vec3 H = normalize(L + E);
// Transform vertex normal into eye coordinates
vec3 N = normalize(modelView * vec4(vNormal, 0.0)).xyz;
// Compute terms in the illumination equation
vec4 ambient = materialAmbient * lightAmbient;
float Kd = max(dot(L, N), 0.0);
vec4 diffuse = Kd * materialDiffuse * lightDiffuse;
float Ks = pow(max(dot(N, H), 0.0), shininess);
vec4 specular = Ks * materialSpecular * lightSpecular;
if(dot(L, N) < 0.0) specular = vec4(0.0, 0.0, 0.0, 1.0);
gl_Position = projection * modelView * vPosition;
color = ambient + diffuse + specular;
color.a = 1.0;
}
Ok, it's working now. The solution was to replace glUniform3fv with glUniform4fv, I guess because the glsl counterpart is a vec4 instead of a vec3. I thought that it would be able to recognize this and simply add a 1.0 to the end, but no.

pixel perfect texture mapping with modern OpenGL

After deciding to try programming in modern OpenGL, I've left behind the fixed function pipeline and I'm not entirely sure about getting the same functionality I had before.
I'm trying to texture map quads with pixel perfect size, matching the texture size. For example, a 128x128 texture maps to a quad 128x128 in size.
This is my vertex shader.
#version 110
uniform float xpos;
uniform float ypos;
uniform float tw; // texture width in pixels
uniform float th; // texture height in pixels
attribute vec4 position;
varying vec2 texcoord;
void main()
{
mat4 projectionMatrix = mat4( 2.0/600.0, 0.0, 0.0, -1.0,
0.0, 2.0/800.0, 0.0, -1.0,
0.0, 0.0, -1.0, 0.0,
0.0, 0.0, 0.0, 1.0);
gl_Position = position * projectionMatrix;
texcoord = (gl_Position.xy);
}
This is my fragment shader:
#version 110
uniform float fade_factor;
uniform sampler2D textures[1];
varying vec2 texcoord;
void main()
{
gl_FragColor = texture2D(textures[0], texcoord);
}
My vertex data is as such, where w and h are the width and height of the texture.
[
0, 0,
w, 0,
w, h,
0, h
]
I load a 128x128 texture and with these shaders I see the image repeated 4 times: http://i.stack.imgur.com/UY7Ts.jpg
Can anyone offer advice on the correct way to be able to translate and scale given the tw th, xpos, xpos uniforms?
There's a problem with this:
mat4 projectionMatrix = mat4( 2.0/600.0, 0.0, 0.0, -1.0,
0.0, 2.0/800.0, 0.0, -1.0,
0.0, 0.0, -1.0, 0.0,
0.0, 0.0, 0.0, 1.0);
gl_Position = position * projectionMatrix;
Transformation matices are right associative, i.e. you should multiply the opposite order. Also you normally don't specify a projection matrix in the shader, you pass it as a uniform. OpenGL provides you ready to use uniforms for projection and modelview. In OpenGL-3 core you can reuse the uniform names to stay compatible.
// predefined by OpenGL version < 3 core:
#if __VERSION__ < 400
uniform mat4 gl_ProjectionMatrix;
uniform mat4 gl_ModelviewMatrx;
uniform mat4 gl_ModelviewProjectionMatrix; // premultiplied gl_ProjectionMatrix * gl_ModelviewMatrix
uniform mat4 gl_ModelviewInverseTranspose; // needed for transformin normals
attribute vec4 gl_Vertex;
varying vec4 gl_TexCoord[];
#endif
void main()
{
gl_Position = gl_ModelviewProjectionMatrix * gl_Vertex;
}
Next you must understand that texture coordinates don't address texture pixels (texels), but that the texture should be understood as a interpolating function with the given sampling points; texture coordinates 0 or 1 don't hit the texel's centers, but lie exactly between the wraparound, thus blurring. As long as your quad on screen size exactly matches the texture dimensions this is fine. But as soon as you want to show just a subimage things get interesting (I leave it as an exercise to the reader to figure out the exact mapping; hint: You'll have the terms 0.5/dimension and (dimension - 1)/dimension in the solution)