Shoot$\_$Ray function with explanation please

The direction of the reflected has been given in class and can be expressed as r $=2$n$($n$$v$)-$v$.$

Vector$4$d shoot$\_$ray$($const Vector$3$d &ray$\_$origin, const Vector$3$d &ray$\_$direction, int max$\_$bounce$)$

$\{$

$//$Intersection point and normal, these are output of find$\_$nearest$\_$object

Vector$3$d p$,$ N;

const int nearest$\_$object $=$ find$\_$nearest$\_$object$($ray$\_$origin, ray$\_$direction, p$,$ N$)$;

if $($nearest$\_$object $<0)$

$\{$

$//$ Return a transparent color

return Vector$4$d$(0,0,0,0)$;

$\}$

$//$ Ambient light contribution

const Vector$4$d ambient$\_$color $=$ obj$\_$ambient$\_$color.array$()*$ ambient$\_$light.array$()$;

$//$ Punctual lights contribution $($direct lighting$)$

Vector$4$d lights$\_$color$(0,0,0,0)$;

for $($int i $=0$; i $<$ light$\_$positions.size$()$; $++$i$)$

$\{$

const Vector$3$d &light$\_$position $=$ light$\_$positions$[$i$]$;

const Vector$4$d &light$\_$color $=$ light$\_$colors$[$i$]$;

const Vector$3$d Li $=($light$\_$position $-$ p$).$normalized$()$;

$//$ TODO: Shoot a shadow ray to determine if the light should affect the intersection point and call is$\_$light$\_$visible

Vector$4$d diff$\_$color $=$ obj$\_$diffuse$\_$color;

if $($nearest$\_$object $==4)$

$\{$

$//$Compute UV coodinates for the point on the sphere

const double x $=$ p$(0)-$ sphere$\_$centers$[$nearest$\_$object$][0]$;

const double y $=$ p$(1)-$ sphere$\_$centers$[$nearest$\_$object$][1]$;

const double z $=$ p$(2)-$ sphere$\_$centers$[$nearest$\_$object$][2]$;

double tu $=$ acos$($z $/$ sphere$\_$radii$[$nearest$\_$object$])/3\mathrm{.}1415$;

double tv $=(3\mathrm{.}1415+$ atan$2($y$,$ x$))/(2*3\mathrm{.}1415)$;

tu $=$ std::min$($tu$,1\mathrm{.}0)$;

tu $=$ std::max$($tu$,0\mathrm{.}0)$;

tv $=$ std::min$($tv$,1\mathrm{.}0)$;

tv $=$ std::max$($tv$,0\mathrm{.}0)$;

diff$\_$color $=$ procedural$\_$texture$($tu$,$ tv$)$;

$\}$

$//$ TODO: Add shading parameters

$//$ Diffuse contribution

const Vector$4$d diffuse $=$ diff$\_$color $*$ std::max$($Li$.$dot$($N$),0\mathrm{.}0)$;

$//$ Specular contribution, use obj$\_$specular$\_$color

const Vector$4$d specular$(0,0,0,0)$;

$//$ Attenuate lights according to the squared distance to the lights

const Vector$3$d D $=$ light$\_$position $-$ p;

lights$\_$color $+=($diffuse $+$ specular$).$cwiseProduct$($light$\_$color$)/$ D$.$squaredNorm$()$;

$\}$

Vector$4$d refl$\_$color $=$ obj$\_$reflection$\_$color;

if $($nearest$\_$object $==4)$

$\{$

refl$\_$color $=$ Vector$4$d$(0\mathrm{.}5,0\mathrm{.}5,0\mathrm{.}5,0)$;

$\}$

$//$ TODO: Compute the color of the reflected ray and add its contribution to the current point color.

$//$ use refl$\_$color

Vector$4$d reflection$\_$color$(0,0,0,0)$;

$//$ TODO: Compute the color of the refracted ray and add its contribution to the current point color.

$//$ Make sure to check for total internal reflection before shooting a new ray.

Vector$4$d refraction$\_$color$(0,0,0,0)$;

$//$ Rendering equation

Vector$4$d C $=$ ambient$\_$color $+$ lights$\_$color $+$ reflection$\_$color $+$ refraction$\_$color;

$//$Set alpha to $1$

C$(3)=1$;

return C;

$\}$