double Sphere::Intersect$($Vector source, Vector d$)$

$\{$

$//$ Compute the intersection of a ray with a TRANSFORMED sphere

$//$ You can assume that for a transformed object the vectors 'scaling' and 'translation" are defined.

$//$ where $($scaling$.$x$,$scaling.y$,$scaling.z$)$ are the scale factors in x$,$y$,$ and z direction

$//$ and $($translation$.$x$,$translation.y$,$translation.z$)$ is the translation vector

$//$ STEP $1$: Transform the ray, compute the transformed ray start point and direction

Vector s $=($source $-$ translation$).$Scale$(1/$ scaling.x$,1/$ scaling.y$,1/$ scaling.z$)$;

Vector dir $=$ d$.$Scale$(1/$ scaling.x$,1/$ scaling.y$,1/$ scaling.z$)$;

$//$ STEP $2$: Compute intersection of the transformed ray with the sphere

$//$ A$,$ B$,$ and C are the parameters of the quadratic equation for finding the

$//$ ray intersection parameter t $($see "Ray Tracing" lecture notes$)$

float A $=$ dir.Dot$($dir$)$;

float B $=2*$ dir.Dot$($s $-$ center$)$;

float C $=($s $-$ center$).$Dot$($s $-$ center$)-$ radius $*$ radius;

float t $=-1\mathrm{.}0$; $//$ the parameter t for the closest intersection point or ray with the sphere. If no intersection t$=-1\mathrm{.}0$

$//$ BEGIN SOLUTION RAY$-$SPHERE INTERSECTION

$//================================================================$

$//==$ Delete the line $"$t$=-1\mathrm{.}0$;$"$ and insert your solution instead $==$

$//==$ NOTE $1$: If there is no ray$-$shere intersection set t$=-1\mathrm{.}0==$

$//==$ NOTE $2$: Use C notation so that the code runs in Coderunner $==$

$//==$ NOTE $3$: You might want to use the sqrt$()$ function $==$

$//================================================================$

float discriminant $=$ B $*$ B $-4*$ A $*$ C;

if $($discriminant $<0)\{$

t $=-1\mathrm{.}0$;

$\}$

else $\{$

float t$1=(-$B $-$ sqrt$($discriminant$))/(2*$ A$)$;

float t$2=(-$B $+$ sqrt$($discriminant$))/(2*$ A$)$;

if $($t$1<0$ && t$2<0)\{$

t $=-1\mathrm{.}0$;

$\}$

else if $($t$1>=0$ && t$2>=0)\{$

t $=$ fmin$($t$1,$ t$2)$;

$\}$

else $\{$

t $=$ fmax$($t$1,$ t$2)$;

$\}$

$\}$

$//$ END SOLUTION RAY$-$SPHERE INTERSECTION

return t;

$\}$

Vector Sphere::Normal$($Vector p$)$

$\{$

$//$ STEP $3$: Calculate the correct normal for the transformed sphere

$//$ replace the existing code below and use the values 'scaling' and 'translation" to compute the normal of the transformed sphere

Vector n $=($p $-$ center$).$Scale$(1/$ scaling.x$,1/$ scaling.y$,1/$ scaling.z$).$Normalize$()$;

return n;

$\}$