Fragment shader<\/strong>: Only color (no shading, since that was enough for me).<\/li>\n<\/ul>\n\n\n\nI drew the curve and control points to figure out a good sampling frequency and length.<\/p>\n\n\n\nThen, I created a point and moved it along the curve.<\/p>\n\n\n\n<\/video><\/figure>\n\n\n\nLater, I increased the speed and added more random curves for the point to follow.<\/p>\n\n\n\n<\/video><\/figure>\n\n\n\nAt this point I realized that somehow I need to dictate that the curves are long enough to be visually pleasant by paying attention to the constraints:<\/p>\n\n\n\n
\nMin\/max values based on the scene.<\/li>\n\n\n\n C1 continuity<\/strong> (so the curves connect smoothly).<\/li>\n<\/ul>\n\n\n\nI also drew the tangent of the curve to visualize it. I calculated the tangent by subtracting the last point from the current point (With a little trick for the first point of course).<\/p>\n\n\n\n<\/video><\/figure>\n\n\n\nMoving the Bunny<\/h2>\n\n\n\n Next, I replaced the green point with the given bunny object. I removed all pitch and roll movements at first.<\/p>\n\n\n\n
\nUsed (0, 1, 0)<\/code> as the initial up vector.<\/li>\n\n\n\nUsed the current tangent as the gaze vector.<\/li>\n\n\n\n Created a basis for the tangent and transformed the bunny into this basis.<\/li>\n<\/ul>\n\n\n\nThe problem I encountered at this point was the alignment of the object because there was not a middle or center point of the object. To solve this problem I calculated the middle point by taking average of edge coordinates (highest and lowest etc) and changed the transition matrix based on the object. It worked for the bunny but I knew that this was not a good solution. However I decided to create my object by knowing the center point instead of trying to generate an algorithm or rules for the center point.<\/p>\n\n\n\n<\/video><\/figure>\n\n\n\nAdding Roll Motion<\/h3>\n\n\n\n I introduced a quaternion to make the bunny roll 45 degrees back and forth. I reordered the transformations:<\/p>\n\n\n\n
\nScaling<\/li>\n\n\n\n Rotating<\/li>\n\n\n\n Rolling<\/li>\n\n\n\n Translation<\/li>\n<\/ol>\n\n\n\nThis looked better. Here the initial up vector is looking down. At this stage I did not realize this mistake. But towards the end of the homework I solved the issue. It was related to the vector calculations.<\/p>\n\n\n\n<\/video><\/figure>\n\n\n\nProblems I Noticed at This Stage<\/p>\n\n\n\n
\nFully random curves sometimes looked weird. The control point generator needed tuning.<\/li>\n\n\n\n The object\u2019s movement speed and rolling speed needed adjustment for smoother motion.<\/li>\n<\/ul>\n\n\n\nMoving to Bezier Surfaces<\/h2>\n\n\n\n I wrote an Octave script to visualize Bezier surfaces. I colored points differently to make design easier.<\/p>\n\n\n\nHalf-Sphere Example<\/h3>\n\n\n\n I created a customizable half-sphere-like Bezier surface generator with Octave. With the code half sphere like Bezier surfaces can be generated by defining the center point and some parameters based on the center point ( which I defined by experimenting with them and which I can not interpret exactly what they are doing \ud83d\ude43.)<\/p>\n\n\n\nThe Octave code:<\/p>\n\n\n\n
function [P] = bezierHalfConvex(r, s, u, k, b)\n P = { [r 0 0], [r r*s 0], [-r r*s 0], [-r 0 0]; ...\n [(r*u) 0 b], [(r*u) (r*s*u) k], [(-r*u) (r*s*u) k], [(-r*u) 0 b]; ...\n [(r*u) 0 b], [(r*u) -(r*s*u) k], [(-r*u) -(r*s*u) k], [(-r*u) 0 b]; ...\n [r 0 0], [r -r*s 0], [-r -r*s 0], [-r 0 0] \n };\nend<\/code><\/pre>\n\n\n\nI also wrote Octave functions to rotate and translate the surfaces. All Octave codes are uploaded with the homework.<\/p>\n\n\n\n
<\/p>\n\n\n\n
Creating the Moving Object<\/h3>\n\n\n\n With the Bezier surface calculator, half sphere generator, rotation, translation codes in my hand, I started to create an object by using Bezier Surfaces. The definitions can be found in <\/mark>bezierScript.m<\/mark> file. I tried to draw something like this:<\/p>\n\n\n\nThis is what I could before getting bored by point calculations. <\/p>\n\n\n\n
\ud83c\udf7d\ufe0f Ingredients are:<\/p>\n\n\n\n
\n2 Bezier half convexes (I used my script for all half convexes) for the front and back of the plane.<\/li>\n\n\n\n 1 half cylinder for base of the plane<\/li>\n\n\n\n 2 half cylinder like shapes for the sitting area, current shape:\n1 to close sitting area completely<\/li>\n\n\n\n 2 for wings<\/li>\n\n\n\n 1 for the wing like shape on the upper back of the plane, up to now:<\/li>\n<\/ul>\n\n\n\nAdding some cosmetics:<\/p>\n\n\n\n
\n1 half convex for the steering wheel shaft<\/li>\n\n\n\n 2 half convex for the sterring wheel, the end result:<\/li>\n<\/ul>\n\n\n\nMoving to C++<\/h3>\n\n\n\n I transferred the points to C++ by simply outputting the calculated points from Octave script and:<\/p>\n\n\n\n
\nGenerated 30 Bezier curves.<\/li>\n\n\n\n Used those points to create a 30\u00d730 resolution Bezier surface.<\/li>\n\n\n\n Calculated indices using a simple triangle method (3 vertices per triangle \u2192 1682 * 3 indices).<\/li>\n\n\n\n Calculated the normals by averaging the triangle normals. However, I made a mistake at this point. I realized it too late, and I had already spent a lot of time on the homework. My normal calculations are not fully appropriate for the homework specification, since I calculated them using only the current surface. Because of how I structured the code (I kept the surfaces isolated from each other), fixing this would be a laborious task. I thought about manually selecting adjacent surfaces and writing loops for those, but that feels tiring and like an ugly solution. Another option is merging the surfaces and storing them as a whole, and recalculating the normals, but that would require a good amount of refactor. So, I sadly decided to skip this requirement.<\/li>\n<\/ul>\n\n\n\nExporting and Rendering<\/h3>\n\n\n\n\nExported the generated object to an OBJ file to use the existing code for obj files.<\/li>\n\n\n\n Adapted rotation, scale, and translation matrices to put the object on the curve correctly.<\/li>\n<\/ul>\n\n\n\nProblems and Fixes<\/h3>\n\n\n\n\nUpside-Down Object<\/strong>\n\nThere was a mistake in the up vector calculation. It was upside down. I corrected the mistake by simply changing the sign.<\/li>\n<\/ul>\n<\/li>\n\n\n\n Incorrect Normals<\/strong>\n\nSome surfaces were facing the wrong way which causes them to be ignored because of the back face culling.<\/li>\n\n\n\n Quick fix: Reversed the order of Bezier curves in the points data.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\nAssigning the Key Presses<\/h2>\n\n\n\n\nSpace<\/strong>: Pauses movement.<\/li>\n\n\n\nP<\/strong>: Stops drawing the curve and curve related things (the tangent vector, up vector etc.)<\/li>\n\n\n\nW<\/strong>: Enables wireframe mode (I had to modify the shader).<\/li>\n<\/ul>\n\n\n\nAfter pressing Space and P:<\/p>\n\n\n\nAfter pressing Space + P + W:<\/p>\n\n\n\nLast-Minute Changes<\/h3>\n\n\n\n The homework required the Bezier surface resolution (s<\/code> and t<\/code>) to be parametric. I initially thought they were separate but realized they were the same. Instead of reverting, I just set both to the same value. So now both of them are customizable but the program takes one argument for them.<\/p>\n\n\n\nFinal Thoughts<\/h2>\n\n\n\n It was a long but rewarding homework! There are still some imperfections, but I learned a lot. You can see a resolution vs fps table at the end.<\/p>\n\n\n\n<\/video><\/figure>\n\n\n\nBezier Surface Resolution vs FPS Table<\/h3>\n\n\n\n My monitor is 144hz. There is a performance problem about wireframe mode when resolution is high (Up to 96 there is no problem). I do not know the reason. To enable wireframe mode I am<\/p>\n\n\n\n
\nChanging glPolygonMode from “fill” to “line”<\/li>\n\n\n\n Adding an if condition into the vertex shader<\/li>\n<\/ul>\n\n\n\nResolution<\/strong><\/td>Frame per Second<\/strong><\/td><\/tr>3×3<\/td> 144 (constant)<\/td><\/tr> 6×6<\/td> 144 (constant)<\/td><\/tr> 12×12<\/td> 144 (constant)<\/td><\/tr> 24×24<\/td> 144 (constant)<\/td><\/tr> 48×48<\/td> 144 (constant)<\/td><\/tr> 96×96<\/td> 144 (constant)<\/td><\/tr> 200×200<\/td> 144 (constant) 400×400<\/td> 144 (constant) 800×800<\/td> 64-65 (stable) 1600×1600<\/td> 18-19 (stable)<\/p>\n","protected":false},"excerpt":{"rendered":"
Hey everyone! I wanted to share my experience working on my computer graphics homework, where I implemented Bezier curves and surfaces. It was a fun but sometimes frustrating journey. Starting with Bezier Curves First, I took the given 2D Bezier Curve MATLAB program and extended it to 3D. I wrote a function to generate Bezier […]<\/p>\n","protected":false},"author":8372,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[1],"tags":[],"class_list":["post-4","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/blog.metu.edu.tr\/e251884\/wp-json\/wp\/v2\/posts\/4","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.metu.edu.tr\/e251884\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.metu.edu.tr\/e251884\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/e251884\/wp-json\/wp\/v2\/users\/8372"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/e251884\/wp-json\/wp\/v2\/comments?post=4"}],"version-history":[{"count":0,"href":"https:\/\/blog.metu.edu.tr\/e251884\/wp-json\/wp\/v2\/posts\/4\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.metu.edu.tr\/e251884\/wp-json\/wp\/v2\/media?parent=4"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/e251884\/wp-json\/wp\/v2\/categories?post=4"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/e251884\/wp-json\/wp\/v2\/tags?post=4"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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