2.5 Curves and Surfaces
2.5.1 2D Implicit Curves
implicit equation $f(x,y)=0$
2.5.2 The 2D Gradient
$nabla f(x,y)=(frac{partial f}{partial x},frac{partial f}{partial y})$
normal vector: vector perpendicular to the tangent vector of the curve at that point
gradient points indicate the direction of the f(x,y) > 0 region
- implicit 2D lines
- implicit Quadric Curves
$Ax^2+Bxy+Cy^2+Dx+Ey+F$
2.5.3 3D Implicit Surfaces
$f(x,y,z)=0$
2.5.4 Surface Normal to an Implicit Surface
A surface normal is a vector perpendicular to the surface.
Each point on the surface may havee a different normal vector
$vec n = nabla f(vec p) = (frac{partial f(vec p)}{partial x},
frac{partial f(vec p)}{partial x},
frac{partial f(vec p)}{partial x})$
2.5.5 Implicit Planes
$(vec p- vec a) cdot vec n = 0 $
$vec n = (vec b - vec a)times(vec c - vec a)$
- 3D Quadric Surfaces
- 3D Curves from Implicit Surfaces
A 3D curve can be constructed from the intersection of two simultaneous implicit equation
2.5.6 Parametric Curves
$begin{bmatrix} xy end{bmatrix} = begin{bmatrix} g(t)h(t) end{bmatrix}$ or $vec p = f(t)$
- 2D Parametric Lines
$begin{bmatrix} xy end{bmatrix} = begin{bmatrix} x_0+t(x_1-x_0)y_0+t(y_1 - y_0) end{bmatrix}$ - 2D Parametric Circles
$begin{bmatrix} xy end{bmatrix} = begin{bmatrix} x_c+rcosphiy_c+rsinphi end{bmatrix}$
2.5.7 3D Parametric Curves
a spiral aroung the z-axis:
$ x=cost $
$ y=sint $
$ z=t $
- 3D Parametric Lines
$ vec p = vec o + tvec d$
2.5.8 3D Parametric Surfaces
$ x = f(u,v) $
$ y = g(u,v) $
$ z = h(u,v) $ - spherical coordinate system
$ x = rcosphi sintheta $
$ y = rsinphi sintheta $
$ z = rcostheta $
2.5.9 Summary of Curves and Surfaces
- implicit curves in 2D / surfaces in 3D
scalar valued functions: $f: mathbb{R}^2 to mathbb{R}$ or $f: mathbb{R}^3 to mathbb{R}$
$S = {mathbf{p} | f(mathbf{p}) = 0 }$ - Parametric curves in 2D / 3D
vector valued functions of one variable $mathbf{p}: D subset mathbb{R} to mathbb{R^2}$ or $mathbf{p}: D subset mathbb{R} to mathbb{R^3}$
$S = { mathbf{p}(t) | t in D }$ - Parametric surfaces in 3D
vecotr valued functions of two variables
$mathbf{p}: subset mathbb{R^2} to mathbb{R^3}$
$S = { mathbf{p}(t) | (u,v) in D }$
2.6 Linear Interpolation
points are connected by straight line segments
$f(x) = y_{i+1} + frac{x-xi}{x{i+1}-xi}(y{i+1}-y_i)$
$(1-t)A+B$
2.7 Triangles
2.7.1 2D Traingles
area=$frac{1}{2} begin{vmatrix} x_b-x_a & x_c - x_a y_b-y_a&y_c-y_a end{vmatrix}$
This area will have a positive sign if points a,b,c are in counterclockwise order and a negative sign,otherwise
-
barycentric coordinate
$mathbf{p} = mathbf{a} + beta(mathbf{b}-mathbf{a}) + gamma(mathbf{c} - mathbf{a}) $
$mathbf{p} = (1-beta-gamma)mathbf{a} + beta mathbf{b} + gamma mathbf{c} $
$mathbf{p} = alpha mathbf{a} + beta mathbf{b} + gamma mathbf{c}$ $alpha+beta+gamma=1$A particular nice feature of barycentric coordinates is that a point p is insde the triangle formed by a,b,c if and only if
$0 < alpha < 1$
$0 < beta < 1$
$0 < gamma < 1$
if one of the coordinates is zero and the other two are between zero and one, then you are on an edge, If two of the coordinate are zeor, then the other is one, and you are at a vertex -
how to compute?
$begin{bmatrix} x_b-x_a&x_c-x_a y_b-y_a&y_c-y_aend{bmatrix} begin{bmatrix}beta\gammaend{bmatrix}=begin{bmatrix}x_p-x_a y_p-yaend{bmatrix}$
$beta = frac{f{ac}(x,y)}{f_ac{x_b,y_b}}$
$beta = frac{(y_a-y_b)x + (x_b-x_a)y + x_ay_b - x_by_a}{(y_a-y_b)x_c + (x_b-x_a)y_c + x_ay_b - x_by_a}$
$A=A_a+A_b+A_c$ is the area of the traingle
$alpha=A_a/A$
$beta=A_b/A$
$gamma=A_c/A$This rule still holds for points outside the triangle if the areas are allowed to be signed. note that these are signed areas and will be computed correctly as long as the same signed area computation is used for both A and subtriangles
2.7.2 3D Triangles
$mathbf{p} = (1-beta-gamma)mathbf{a} + betamathbf{b} + gammamathbf{c}$
$mathbf{n=(b-a)times(c-a)}$
$area=mathbf{frac{1}{2}|(b-a)times(c-a)|}$
$alpha=mathbf{ frac{n cdot n_a}{|n|^2}}$
$beta=mathbf{ frac{n cdot n_b}{|n|^2}}$
$gamma=mathbf{ frac{n cdot n_c}{|n|^2}}$
近期评论