Surface Modeling and Bilinear Surface

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written 5.7 years ago by

tomarsakshi32 • 90

The values for x,y and z coordinates can be determined as,

$\begin{equation} P_x(u,v)=10(1-u-v)+8(u)+8(v)\ =10-10u-10v+8u+8v\ =10-2u-2v\ \end{equation}$

$\begin{equation} P_y(u,v)=-8(1-u-v)+4(u)+4(v)\ =-8+8u+8v+4u+4v\ =-8+12u+12v\ \end{equation}$

$\begin{equation} P_z(u,v)=3(1-u-v)+3.2(u)+3.2(v)\ =3-3u-3v+3.2u+3.2v\ =3+0.2u+0.2v\ \ \end{equation}$

Bilinear Section The bilinear surface is the simplest non-flat (curved) surface because it is fully defined by means of its 4 corner points. It's four boundary curves are straight lines and the coordinates of any point on this surface are derived by the linear interruption.\ \Let the corner points be the four distinct points $P_0{}_0$ , $P_0{}_1$ , $P_1{}_0$ , $P_1{}_1$ . The top and bottom boundary curves are straight lines, they are,

$\begin{equation} P(u,0)=(P_1{}_0-P_0{}_0)u+P_0{}_0\=P_0{}_0(1-u)+uP_1{}_0 \end{equation}$

$\begin{equation} P(u,1)=(P_1{}_1-P_0{}_1)u+P_0{}_1\ =P_0{}_1(1-u)+uP_1{}_1\ \end{equation}$ To generate a curve, we calculate interpolation function using parameters V.\ For top boundary V=0 and for bottom boundary V=1. Surface is created by linear interpolating between top and bottom boundary.\ Resulting Surface equation is,

$\begin{equation} P(u,v)=P(u,0)(i-v)+vP(u,1)\ =(1-v)[(1-u)P_0{}_0+uP_1{}_0]\ =(1-v)[(1-u)P_0{}_0+uP_1{}_0]+v[(1-u)P_0{}_1+uP_1{}_1]\ \end{equation}$

$\begin{equation} P_0{}_1+uvP_1{}_1\ =[(1-u) u]\ \end{equation}$

$\begin{bmatrix} (1-u) & u \end{bmatrix}$

$\begin{bmatrix} P_0{}_0 & P_0{}_1 \\ P_1{}_0 & P_1{}_1 \\ \end{bmatrix}$

$\begin{bmatrix} (1-v)\\ v\\ \end{bmatrix}$ Where,\ 0

$\leq$ u

$\leq$ 1 and \ 0

$\leq$ v

$\leq$ 1

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