Overview
++ In this project, I explored the world of mesh editing through building Bezier curves and surfaces using the + de Casteljau algorithm and implementing various mesh operations such as area-weighted vertex normals, edge + flip, edge split, and loop subdivision. +
++ +
Section I: Bezier Curves and Surfaces
+ +Part 1: Bezier Curves with 1D de Casteljau Subdivision
+ +
+
+ Briefly explain de Casteljau's algorithm and how you implemented it in order to evaluate Bezier curves.
+
+
+
+ +
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+
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+ de Casteljau's Algorithm takes in a set of control points and a parameter
t, a + proportion + of length along the line and evaluates a + Bezier curve by recursively interpolating between each pair of control points. It can repeat this + process until the criterion has been met or that the final interpolated point has been calculated. + By + adjusting this parametert, it can find all the points along the curve. I implemented + this + algorithm by looping through each point and its adjacent point, \(p_i\) and \(p_{i+1}\), and + computing the interpolated point \(p_i^{'} = \text{lerp}(p_i, p_{i + 1}, t) = (1 - t) p_i + t p_{i + + 1}\). After each iteration, there will be one fewer control point than the previous iteration. This + process repeats until there is only one point left, which is the final evaluated point. +
+
+ +
+
+ Take a look at the provided
+ .bzc files and create your own Bezier curve with 6 control
+ points of your choosing. Use this Bezier curve for your screenshots below.
+
+ +
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+
+
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+ Here is a Bezier curve with 6 control points of my choosing:
+ ++
+
++ ++ +
+ Bezier Curve +
+
+
+
+ Show screenshots of each step / level of the evaluation from the original control points down to the final
+ evaluated point. Press E to step through. Toggle C to show the completed Bezier curve
+ as well.
+
+ +
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+
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+ Here are the screenshots of each step of the evaluation from the original control points down to the
+ final evaluated point as well as the completed Bezier curve:
+ ++
+
++ ++ +
+ Step 0 ++ +
+ Step 1 ++ ++ +
+ Step 2 ++ +
+ Step 3 ++ ++ +
+ Step 4 ++ +
+ Step 5 +
++
++ ++ +
+ Completed Bezier Curve +
+
+
+
+ Show a screenshot of a slightly different Bezier curve by moving the original control points around and
+ modifying the parameter \(t\) via mouse scrolling.
+
+ I had shifted \(t\) to a higher value which meant that the curve was more towards the right. I also
+ moved the control points around to create a different curve. Here is a screenshot of a slightly
+ different Bezier curve by moving the original control points around
+ and modifying the parameter \(t\) via mouse scrolling:
+
+
+
+
+ + +
+
+
+
+ |
+
+ |
+
+ 
+ |
+
+ + +
Part 2: Bezier Surfaces with Separable 1D de Casteljau
+
+ Briefly explain how de Casteljau algorithm extends to Bezier surfaces and how you implemented it in order to
+ evaluate Bezier surfaces.
+
+ +
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+
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+ A 3D Bezier surface is an \(n \times n\) grid of control points where there are \(n\) parallel
+ Bezier
+ curves in \(u\). The separable 1D de Casteljau's algorithm can evaluate the surface position
+ corresponding to \(u, v\) along an axis \(x\) and an orthogonal axis \(y\). This algorithm extends
+ by
+ first finding the final interpolated point \(u\) at each of these \(n\) Bezier curves. Each of these
+ points combined will help make up a new set of \(n\) control points for the "moving" Bezier curve.
+ Finally, the 1D de
+ Casteljau's algorithm can evaluate \(v\) on this final curve. I implemented this algorithm by
+ first evaluating the \(n\) parallel Bezier curves in \(u\) and storing them into a new
+
vector. The resulting \(n\) points became my next set of control pointers for another + Bezier curve in \(v\). This process repeats until the final point is + evaluated. +
+
+
+ Show a screenshot of
+
+ bez/teapot.bez (not code) +
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+
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+ Here is a screenshot of
bez/teapot.bezevaluated by my implementation of the Bezier + surface: ++++
++ ++ +
+ Bezier Surface of a Teapot +
+
+
+
+ + +
Section II: Triangle Meshes and Half-Edge Data Structure
+ +Part 3: Area-Weighted Vertex Normals
+
+ Briefly explain how you implemented the area-weighted vertex normals.
+
+ +
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+
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+ I implemented the area-weighted vertex normals by making a constant iterator of the half-edge data
+ structure to traverse over all of the
+ neighboring triangles and weighting each one by its area. I defined a
find_area+ function + that used the cross product formula of the vertices to find the area of the triangle. + Here are the + formal steps I took to + implement the area-weighted vertex normals: +-
+
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+ I initialized an empty
Vertex3D vertexto keep track of the weighted vertex. +
+ -
+ I found the starting half-edge and used a
do-whileloop to traverse through all + the + triangles and stopping once we reached the original initial half-edge. +
+ -
+ For each triangle, I calculated the area of the triangle using the cross product formula.
+ This
+ function found all three vertices of the triangle by using the
nextand +vertexmethods. I then found the difference vectors and took the cross product + before normalizing the result and dividing by 2 because the area of a triangle is half the + area + of the parallelogram formed by the vectors. +
+ - + I used this calculated area to weight the normal of the triangle and added it to the + weighted + vertex from earlier. + +
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+ I called on the
twin().next()to find the next half-edge and face. +
+ -
+ Finally, once all of the half-edges have been traversed, I normalized the weighted vertex by
+ calling
unit()on it. +
+
+ -
+ I initialized an empty
+
+ Show screenshots of
+
+ dae/teapot.dae (not .bez) comparing teapot shading with and
+ without vertex normals. Use Q to toggle default flat shading and Phong shading.
+ -
+
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+ Here are some screenshots of
dae/teapot.daeshading with and without vertex normals: ++++
++ ++ +
+ Mesh without Vertex Normals ++ +
+ Mesh with Phong Shading ++ ++ +
+ No Mesh without Vertex Normals ++ +
+ No Mesh with Phong Shading +
+
+
+ +
Part 4: Edge Flip
+ +
+ Briefly explain how you implemented the edge flip operation and describe any interesting implementation /
+ debugging tricks you have used.
+
+ +
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+
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+ I first started by creating a diagram of each of the half-edges, edges, vertices and faces before
+ and
+ after the flip to ensure that the pointers would be correct. Here is the diagram shown below:
+ ++
+
++ ++ +
+ Before and After Flip Diagram +
+
+ -
+ Here are the formal steps I took to implement the edge flip operation:
+
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+
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+ First, I checked if
e0->isBoundary()wastrueto make sure to + never + flip a boundary edge and simply returned if it was. +
+ -
+ Then, I defined the inner and outer half-edges of the two triangles using the
+
twin()andnext()methods. Each of these half-edges corresponded + to + the 10 half-edges,h0 ... h9, as shown in the diagram above. +
+ -
+ Next, I defined the vertices of the two triangles using the
vertex()method on + the + appropriate half-edge. Each of these vertices corresponded to the 4 vertices, +v0 ... v3, as shown in the diagram above. +
+ -
+ Afterwards, I defined the edges and faces of the two triangles using the
edge()+ and +face()methods on the appropriate half-edge. Each of these edges and faces + corresponded to the 5 edges,e0 ... e4, and 2 faces,f0, f1, as + shown + in + the diagram above. +
+ -
+ Then, I updated each of the 10 half-edge pointers using the
setNeighbors()+ method + according to the diagram above. +
+ -
+ Finally, I reassigned the half-edge pointers for each of the 4 vertices, 5 edges, and 2
+ faces
+ according to the diagram above and returned the newly updated
e0. +
+
+ -
+ First, I checked if
+ +
+
+ Show screenshots of the teapot before and after some edge flips.
+
+
+ -
+
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+ Here are some screenshots of
dae/teapot.daebefore and after some edge flips. ++++
++ ++ +
+ Before Edge Flips ++ +
+ After Edge Flips +
+
+
+ +
+ Write about your eventful debugging journey, if you have experienced one.
+
+
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+
- + In the process of implementing the edge flip operation, I ran into some issues where the mesh would + look + a bit distorted and that edges would disappear after the flip. I realized that in my diagram of the + half-edge data structure, I had not taken into account the half-edge pointers for the edges on the + outside of the current mesh element. As a result, I went back to my implementation and made sure to + redraw the half-edge data structure to include the outer edges and vertices and correctly updated + the + pointers. Afterwards, the mesh looked much better after each of the edge flips. + +
- + Here is the incorrect half-edge flip: + +
- + + +
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+ Here are some screenshots of
dae/teapot.daebefore and after some edge flips. ++++
++ ++ +
+ Before Edge Splits ++ +
+ After Edge Splits +
+
+ -
+ Here are some screenshots of
dae/teapot.daebefore and after some edge flips. ++++
++ ++ +
+ Before Edge Flips and Splits ++ +
+ After Edge Flips and Splits +
+
+
+
+
+
+
+
+
+
+ 
+ |
+
+
+
Part 5: Edge Split
+
+ Briefly explain how you implemented the edge split operation and describe any interesting implementation
+ /
+ debugging tricks you have used.
+
+ +
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+
+ Show screenshots of a mesh before and after some edge splits.
+
+ +
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+
+ +
+ Show screenshots of a mesh before and after a combination of both edge splits and edge flips.
+
+ +
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+
+ +
+ Write about your eventful debugging journey, if you have experienced one.
+
+
+
+ + YOUR RESPONSE GOES HERE +
+
Part 6: Loop Subdivision for Mesh Upsampling
+ + Briefly explain how you implemented the loop subdivision and describe any interesting implementation / + debugging tricks you have used. + ++ YOUR RESPONSE GOES HERE +
++ + + + Take some notes, as well as some screenshots, of your observations on how meshes behave after loop + subdivision. What happens to sharp corners and edges? Can you reduce this effect by pre-splitting some + edges? + +
+ YOUR RESPONSE GOES HERE +
++ + + + Load dae/cube.dae. Perform several iterations of loop subdivision on the cube. Notice that the cube + becomes + slightly asymmetric after repeated subdivisions. Can you pre-process the cube with edge flips and splits + so + that the cube subdivides symmetrically? Document these effects and explain why they occur. Also explain + how + your pre-processing helps alleviate the effects. + +
+ YOUR RESPONSE GOES HERE +
++ + + + If you have implemented any extra credit extensions, explain what you did and document how they work + with + screenshots. + +
+ YOUR RESPONSE GOES HERE +
++ + +
Part 7 (Optional, Possible Extra Credit)
+ + Save your best polygon mesh as partsevenmodel.dae in your docs folder and show us a screenshot of the + mesh + in your write-up. + ++ YOUR RESPONSE GOES HERE +
++ + + + Include a series of screenshots showing your original mesh and your mesh after one and two rounds of + subdivision. If you have used custom shaders, include screenshots of your mesh with those shaders + applied as + well. + +
+ YOUR RESPONSE GOES HERE +
++ + + + Describe what you have done to enhance your mesh beyond the simple humanoid mesh described in the + tutorial. + +
+ YOUR RESPONSE GOES HERE +
++ + + + \ No newline at end of file