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Ordinary Differential Equations $y'(x) = f(x,y)$ Euler's Method > extremely simple approximation with low accuracy > Suppose tha twe know an initial state $y(x_0) = y_0$ > Approximate $y$ at $x = x_0 + h$, $h$ is a small step size $\frac{y(x_0+h) -y_0}{h} = f(x,y)$ => $y(x_0 +h) = y_0 + h f(x,y)$ - General formula > $x_{i+1} = x_i +h$ > $y_{i+1} = y_i + h f(x_i, y_i)$ Talyor Series - Approximati..

Topics - Solution of Nonlinear Equations > Bracketing Methods > Bisection Method - Open Methods > Newton-Raphson Method > Secant Method Basis of Bisectio Method - An equation $f(x) =0$, where $f(x)$ is real continuous function, has at least one root between $x_1$ and $x_u$ if $f(x_1) f(x_u) roots of the equation may still exist between the two points > there may not be any roots for the equation..

Optimization - Golden Ratio Search - Fibonacci Search - Methods for locating extrema (maxima or minima) of functions Functions of One Variable - The function $f$ has a local minimum value at $x=p$, if there exist an open interval $I$ containing $p$ s.t. $f(p) \leq f(x) $ for all $x \in I$ - $f$ has a local maximun value at $x=p$, if there exists an open interval $I$ containing $p$ s.t. $f(x) \le..

Least-Squares Fitting - Least-Squares Line - Power Fit - Data Linearization - Linear Least-Squares - Nonlinear Least-Squares Least-Squares Line - Data points $(x_1, y_1), \cdots, (x_N, y_N), {x_k}$ are distinct - Numerical method is to defermine $y= f(x)$ - Linear approximation : $y = f(x) = Ax + B$ - if there is an error in the data points : $f(x_k) = y_k + e_k$, where $e_k = f(x_k) - y_k$ is t..

Interpolation Interpolation - A process of finding a function that passes through a given set of points $(x_i, y_i)$ (data fitting or curve fitting) - it can be used to estimate variable $y$ corresponding to unknown $x \in [a,b]-{x_i}$ Extrapolation - Estimate variable $y$ corresponding to $x x_n$ Interpolation EX $x_0 = 0, x_1 = \frac{\pi }{4}, x_2 = \frac{\pi }{2}$ $y_i = \cos x_i..

Numerical Methods - Jacobi Method - Causs-Seidel Method Solving Large Linear Systems - Global illumination (Radiosity) : not only direct illumination but also indirect illumination ( comming to other surface point not light source) - Solve $B = (I - \rho F)^{-1}E$ : (E : light energe, B : individual color ) - Need to solve for a large matrix - it takes too much time with conventional methods Jac..

Numerical Methods - Traiangular Systems Triangular Systems Lower triangular matrix $L$ - Square matrix for which $L_{ij} = $ when $ i j$ Linear system $Ux = r$ - Backward substitution - $x_i = \frac{1}{U_{ii}} ( r_i - \su..

Bounding Volume Construction - Principal Component Analysis (PCA) Principal Components - All principal components (PCs) start at the origin - First PC is direction of maximum variance from origin - Subsequent PCs are orthogonal to 1st PC and describe maximun residual variance - For an arbitrary set of $N$ vertices $P_1, $P_2, \cdots , P_N$ - Mean position : $m = \frac{1}{N} \sum_{i=1}^N P_i$ - 3..

Rendering Rendering Curves - Evaluate $p(u)$ for serveral values of $u$, and draw line segments between these points. - The finer the divisions of the curve (the more values of $u$), the better the rendering will look. - For efficiency : $p(u) = c_0 + uc_1 + u^2 c_2 + u^3c_3 = c_0 + u(c_1 + u(c_2 uc_3)) $ - Smaller number of point in flat curve Subdivision of ezier Curves - Divide a curve into 2..

Surface - Bilinear surfaces - Bi-cubic surfaces - Bezier surfaces - Spline surfaces - Rendfering surfaces To Represent boundaries of objects exactly Implicit Surfaces - in 3D, surfaces are represended as : $f(x,y,z) = 0$ Parametric Surfaces - The value of each component $(x,y,z)$ depends on independent variables, $u$ and $v$. - $p(u,v) = \begin{bmatrix} x(u,y) \\ y(u,v) \\ z(u,v)\end{bmatrix}$ P..