Chapter 7: Conclusions and Future Works
7.2 Future Works
The hair model consumes a lot of editing time. It would be a research direction for editing the hair model interactively to achieve the feature of “what we see is what we get”. The interpolation scheme has drawbacks. It could make hairs interpolated from the same key hair look too similar and tufts of hairs look too different from each other when the key hairs are too different.
In the future, we will add the Eulerian fluid to simulate the wind interacting with the hair like [1]. We will add LOD support to the simulation program.
We would like to investigate the possibility of adding the environment lighting [17][29] support to the renderer. Currently, if we add a skybox as the background, the lighting of the hair doesn‟t blend into the scene because the light source doesn‟t match the environment image.
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Appendix A
We give an informal explanation of why making the velocities divergence free
preserves the incompressible feature of fluid here.
Figure 50: An arbitrary small region of fluid
Consider the case shown in Figure 50. The box is an arbitrary small region of fluid,
∂Ω is its boundary surface, n̂ is the surface normal vector and u⃗ is the velocity. The change of volume with respect to time is:
d
dtvolume = ∯ u⃗ ⋅ n̂
∂Ω
(6) If the volume doesn‟t change, we get:
∯ u⃗ ⋅ n̂ integrand must equal to zero. Hence we have:
∇ ∙ u⃗ = 0 (10)
We see that the incompressibility condition is equivalent to making the velocities divergence free.
Appendix B
We are going to start with the split incompressible fluid equations without viscosity, see [11]. In these equations, „q‟ is a generic quantity like velocity, density or temperature, u⃗ is the velocity of the fluid, g⃗ is gravity, ρ is the density of the fluid and p is pressure.
Assume
∂u⃗
∂t =u⃗ n+1− u⃗ n
∆t (14)
We substitute the derivative of velocity into the pressure equation:
u⃗ n+1− u⃗ n
∆t +1
ρ∇p = 0 (15)
We can rearrange the equation to get:
u⃗ n+1− u⃗ n+∆t
ρ ∇p = 0 (16)
We take the gradient on both side of the equation and rearrange it:
∇ ∙ u⃗ n+1− ∇ ∙ u⃗ n+∆t that makes the velocity field divergence free.
On the staggered MAC grid, we use central difference to approximate the divergence of velocity in a fluid cell (i,j,k) :
Here ∆x is the width of the grid cell.
Similarly, we approximate the Laplacian of pressure with:
∇2p =6pi,j,k− pi+1,j,k− pi,j+1,k− pi,j,k+1− pi−1,j,k− pi,j−1,k− pi,j,k−1
∆x2 (20)
Then we have the pressure equation in a fluid cell (i, j, k):
−∆t
If cell (i+1, j, k) is air, we assume it is a free boundary with zero pressure, and then we set the term pi+i,j,k to zero:
We can arrange the system of linear equations of pressure into a matrix form A ∙ p = r:
( gradient of pressure to obtain the divergence free velocities, this term appears on both side of the equation and could be cancelled.
Appendix C
We list the tessellate control/evaluation shader code of the rendering pipeline in step (1) of the rendering stage in Figure 51 and Figure 52.
Figure 51: Tessellate control shader code of the B-spline tessellation step 1 layout(vertices = 4) out;
2 in vec3 vPosition[];
3 patch out vec3 tcTangent[3];
4 uniform vec2 tessLevelOuter;
5 #define ID gl_InvocationID 6 void main()
7 {
8 gl_out[ID].gl_Position = vec4(vPosition[ID], 1);
9 tcTangent[0] = vPosition[1]-vPosition[0];
10 tcTangent[1] = normalize(vPosition[2]-vPosition[1]);
11 tcTangent[2] = vPosition[3]-vPosition[2];
12 gl_TessLevelOuter[0] = tessLevelOuter[0];
13 gl_TessLevelOuter[1] = tessLevelOuter[1];
14 }
Figure 52: Tessellate evaluation shader of the B-spline tessellation step 1 layout(isolines, equal_spacing) in;
2 patch in vec3 tcTangent[3];
3 precise out vec3 teTangent;
4 void main() 5 {
6 float u = gl_TessCoord.x, v = gl_TessCoord.y;
7 vec4 p0 = gl_in[0].gl_Position, p1 = gl_in[1].gl_Position;
8 vec4 p2 = gl_in[2].gl_Position, p3 = gl_in[3].gl_Position;
9 float uu = u*u, uuu = uu*u;
10 float b0 = (1.0-u)*(1.0-u)*(1.0-u)/6.0;
11 float b1 = (3.0*uuu - 6.0*uu +4.0)/6.0;
12 float b2 = (-3.0*uuu + 3.*uu + 3.*u + 1.0)/6.0;
13 float b3 = (uuu)/6.0;
14 precise vec4 outPos =
15 fma(p0,vec4(b0),p1*b1) + fma(p3,vec4(b3), p2*b2);
16 gl_Position = outPos;
17 float Bt[3];
18 Bt[0] = 0.5f*uu - u + 0.5f;
19 Bt[1] = -uu + u + 0.5f;
20 Bt[2] = 0.5f*uu + 0 + 0;
21 teTangent = tcTangent[2]*Bt[2] +
22 fma(tcTangent[0],vec3(Bt[0]),tcTangent[1]*Bt[1]);
23 }
We list the tessellate control/evaluation shader code of the rendering pipeline in step (2) of the rendering stage in Figure 53 and Figure 54.
Figure 53: Tessellate control shader code of single strand interpolation step 1 layout(vertices = 1) out;
2 uniform vec2 tessLevelOuter;
3 in int vVertexID[];
4 patch out int tcVertexID;
5 void main() 6 {
7 tcVertexID = vVertexID[0];
8 gl_TessLevelOuter[0] = tessLevelOuter[0];
9 gl_TessLevelOuter[1] = tessLevelOuter[1];
10 }
Figure 54: Tessellate evaluation shader code of single strand interpolation step
11 uniform int numSegmentPerHair;
12 patch in int tcVertexID; //<-workaround: gl_PrimitiveID missing 13 out vec3 teTangent;
20 int hairID = tcVertexID / (numSegmentPerHair);
21 int vertexIndex = 2*tcVertexID + vertexID;
22 vec2 coord = texelFetch( clumpCoord, interpHairID ).xy;
23 vec3 yAxis = texelFetch( coordFrame, hairID*2 ).xyz;
24 vec3 zAxis = texelFetch( coordFrame, hairID*2 +1).xyz;
25 vec3 offset = yAxis * coord.x + zAxis *coord.y;
26 int vertexID2Root = vertexID+tcVertexID%(numSegmentPerHair);
27 float ratio = float(vertexID2Root)/float(numSegmentPerHair);
28 offset *= (clumpWidth* (rootWidth *(1.0-ratio) + tipWidth * ratio) );
29 vec3 vertPos = texelFetch(keyHairPos, vertexIndex).xyz;
30 vertPos += offset;
31 gl_Position = vec4(vertPos, 1.0);
32 teTangent = texelFetch(hairTangent, vertexIndex).xyz;
33 }