Computer Aided Geometry Design

MATH6110P: Computer Aided Geometric Design (Autumn-Winter 2021-2022)

Instructor: Renjie Chen (renjiec@ustc.edu.cn)

Webpage: http://staff.ustc.edu.cn/~renjiec/Courses/CAGD_2021S1/default.htm

Github Repository: https://github.com/Chaphlagical/CAGD

Slides

• Introduction
• Linear Algebra
• Data Fitting
• Bézier Curve
• Bézier Curve B
• Curve Differential Geometry

Assignments

• Homework#1
• Homework#2
• Homework#3

Report

• Report#1
• Report#2
• Report#3

Release

• Windows 10
• Visual Studio 2019 or download redistributable

Download: Link

Demo

Data Interpolation $$ \begin{pmatrix} \varphi_1(x_1) & \varphi_2(x_1) & \cdots & \varphi_n(x_1)\\ \varphi_1(x_2) & \varphi_2(x_2) & \cdots & \varphi_n(x_2)\\ \vdots & \vdots & \ddots & \vdots\\ \varphi_1(x_n) & \varphi_2(x_n) & \cdots & \varphi_n(x_n) \end{pmatrix} \begin{pmatrix} \alpha_1\\\alpha_2\\\vdots\\\alpha_n \end{pmatrix}= \begin{pmatrix} y_1\\y_2\\\vdots\\y_n \end{pmatrix} $$

Bézier Curve(De Casteljau Algorithm) $$ \begin{aligned} \begin{matrix} \pmb b^0_i(t)=\pmb b_i,&i=0,\cdots,n \end{matrix}\\ \pmb b_i^r(t)=(1-t)\pmb b_i^{r-1}(t)+t\pmb b_{i+1}^{r-1}(t)\\ \begin{matrix} r=1,\cdots,n&i=0,\cdots,n-r \end{matrix} \end{aligned} $$

Bézier Curve(Bernstein Basis) $$ \begin{align} \pmb f(t)&=\sum_{i=0}^nB_i^{(n)}(t)\pmb p_i\\ &=\sum_{i=0}^n\left(\begin{matrix}n\\i\end{matrix} \right)t^i(1-t)^{n-i}\pmb p_i \end{align} $$