Quad
四角形との交差を考えていく.Q Q Q v ⃗ \vec{v} v u ⃗ \vec{u} u
A x + B y + C z = D
\begin{equation}
\begin{split}
Ax+By+Cz=D
\end{split}
\end{equation}
A x + B y + C z = D
これをn ⃗ = ( A , B , C ) , w ⃗ = ( x , y , z ) \vec{n}=(A,B,C),\vec{w}=(x,y,z) n = ( A , B , C ) , w = ( x , y , z ) n ⃗ ⋅ w ⃗ = D
\begin{equation}
\begin{split}
\vec{n} \cdot \vec{w}=D
\end{split}
\end{equation}
n ⋅ w = D
このn ⃗ \vec{n} n r ⃗ = o ⃗ + t d ⃗ \vec{r} = \vec{o} +t \vec{d} r = o + t d r ⃗ = w ⃗ \vec{r} = \vec{w} r = w n ⃗ ⋅ ( o ⃗ + t d ⃗ ) = D
\begin{equation}
\begin{split}
\vec{n} \cdot (\vec{o}+t\vec{d})=D
\end{split}
\end{equation}
n ⋅ ( o + t d ) = D
さて、変換をしていこう.n ⃗ ⋅ ( o ⃗ + t d ⃗ ) = D n ⃗ ⋅ o ⃗ + t n ⃗ ⋅ d ⃗ = D t n ⃗ ⋅ d ⃗ = D − n ⃗ ⋅ o ⃗ t = D − n ⃗ ⋅ o ⃗ n ⃗ ⋅ d ⃗
\begin{equation}
\begin{split}
& \vec{n} \cdot (\vec{o}+t\vec{d})=D \\
& \vec{n} \cdot \vec{o} + t \vec{n} \cdot \vec{d} = D \\
& t \vec{n} \cdot \vec{d} = D - \vec{n} \cdot \vec{o} \\
& t = \frac{D - \vec{n} \cdot \vec{o}}{\vec{n} \cdot \vec{d}}
\end{split}
\end{equation}
n ⋅ ( o + t d ) = D n ⋅ o + t n ⋅ d = D t n ⋅ d = D − n ⋅ o t = n ⋅ d D − n ⋅ o
次はこのt t t n ⃗ , D \vec{n},D n , D n ⃗ \vec{n} n u ⃗ , v ⃗ \vec{u},\vec{v} u , v n ⃗ = u ⃗ × v ⃗
\begin{equation}
\begin{split}
\vec{n} = \vec{u} \times \vec{v}
\end{split}
\end{equation}
n = u × v
これで法線はOK.D D D w ⃗ \vec{w} w Q Q Q w ⃗ = Q \vec{w}=Q w = Q n ⃗ ⋅ Q = D
\begin{equation}
\begin{split}
\vec{n} \cdot Q = D
\end{split}
\end{equation}
n ⋅ Q = D
あとはこれをコードに落とせばOK.m_q,m_u,m_vの3つを用意.Q Q Q n ⃗ \vec{n} n D D D Quad ( const Vec3 & q , const Vec3 & u , const Vec3 & v )
: m_q ( q )
, m_u ( u )
, m_v ( v )
{
Vec3 n = Cross ( m_u , m_v );
m_normal = n . normalized ();
m_d = Dot ( m_normal , m_q );
}
t t t double denom = Dot ( m_normal , ray . direction . xyz ());
if ( Abs ( denom ) < EPSILON_VALUE ) { return false ; }
double t = ( m_d - Dot ( m_normal , ray . origin . xyz ())) / denom ;
if ( ! interval . Contains ( t )) { return false ; }
Q Q Q Q + u ⃗ Q+\vec{u} Q + u Q + v ⃗ Q+\vec{v} Q + v Q + u ⃗ + v ⃗ Q+\vec{u}+\vec{v} Q + u + v P P P P = Q + α u ⃗ + β v ⃗ 0 < = α < = 1 , 0 < = β < = 1
\begin{equation}
\begin{split}
& P = Q + \alpha \vec{u} + \beta \vec{v} \\
& 0 <= \alpha <=1, 0 <= \beta <=1
\end{split}
\end{equation}
P = Q + α u + β v 0 <= α <= 1 , 0 <= β <= 1
と表せる.p ⃗ \vec{p} p p ⃗ = P − Q = α u ⃗ + β v ⃗
\begin{equation}
\begin{split}
\vec{p} = P-Q = \alpha \vec{u} + \beta \vec{v}
\end{split}
\end{equation}
p = P − Q = α u + β v
そしてこのp ⃗ \vec{p} p u ⃗ \vec{u} u u ⃗ × p ⃗ = u ⃗ × ( α u ⃗ + β v ⃗ ) = α u ⃗ × u ⃗ + β u ⃗ × v ⃗ = β u ⃗ × v ⃗
\begin{equation}
\begin{split}
& \vec{u} \times \vec{p} \\
& = \vec{u} \times (\alpha \vec{u} + \beta \vec{v}) \\
& = \alpha \vec{u} \times \vec{u} + \beta \vec{u} \times \vec{v} \\
& = \beta \vec{u} \times \vec{v}
\end{split}
\end{equation}
u × p = u × ( α u + β v ) = α u × u + β u × v = β u × v
となる.v ⃗ \vec{v} v v ⃗ × p ⃗ = v ⃗ × ( α u ⃗ + β v ⃗ ) = α v ⃗ × u ⃗ + β v ⃗ × v ⃗ = α v ⃗ × u ⃗
\begin{equation}
\begin{split}
& \vec{v} \times \vec{p} \\
& = \vec{v} \times (\alpha \vec{u} + \beta \vec{v}) \\
& = \alpha \vec{v} \times \vec{u} + \beta \vec{v} \times \vec{v} \\
& = \alpha \vec{v} \times \vec{u}
\end{split}
\end{equation}
v × p = v × ( α u + β v ) = α v × u + β v × v = α v × u
さて、この後が妙な方法なんだけど、外積だと移項ができないので、内積にしてしまおうという考えでn ⃗ \vec{n} n n ⃗ ⋅ ( v ⃗ × p ⃗ ) = α n ⃗ ⋅ ( v ⃗ × u ⃗ ) n ⃗ ⋅ ( u ⃗ × p ⃗ ) = β n ⃗ ⋅ ( u ⃗ × v ⃗ )
\begin{equation}
\begin{split}
& \vec{n} \cdot (\vec{v} \times \vec{p}) = \alpha \vec{n} \cdot (\vec{v} \times \vec{u}) \\
& \vec{n} \cdot (\vec{u} \times \vec{p}) = \beta \vec{n} \cdot (\vec{u} \times \vec{v})
\end{split}
\end{equation}
n ⋅ ( v × p ) = α n ⋅ ( v × u ) n ⋅ ( u × p ) = β n ⋅ ( u × v )
内積にしたので塊で移行ができる.α , β \alpha,\beta α , β α = n ⃗ ⋅ ( p ⃗ × v ⃗ ) n ⃗ ⋅ ( u ⃗ × v ⃗ ) β = n ⃗ ⋅ ( u ⃗ × p ⃗ ) n ⃗ ⋅ ( u ⃗ × v ⃗ )
\begin{equation}
\begin{split}
& \alpha = \frac{\vec{n} \cdot (\vec{p} \times \vec{v})}{\vec{n} \cdot (\vec{u} \times \vec{v})} \\
& \beta = \frac{\vec{n} \cdot (\vec{u} \times \vec{p})}{\vec{n} \cdot (\vec{u} \times \vec{v})}
\end{split}
\end{equation}
α = n ⋅ ( u × v ) n ⋅ ( p × v ) β = n ⋅ ( u × v ) n ⋅ ( u × p )
さて、ここで
w ⃗ = n ⃗ n ⃗ ⋅ ( u ⃗ × v ⃗ ) = n ⃗ n ⃗ ⋅ n ⃗
\begin{equation}
\begin{split}
\vec{w} = \frac{\vec{n}}{\vec{n} \cdot (\vec{u} \times \vec{v})} = \frac{\vec{n}}{\vec{n} \cdot \vec{n}}
\end{split}
\end{equation}
w = n ⋅ ( u × v ) n = n ⋅ n n
という定義をしておくと、次のように綺麗にまとまる.α = w ⃗ ⋅ ( p ⃗ × v ⃗ ) β = w ⃗ ⋅ ( u ⃗ × p ⃗ )
\begin{equation}
\begin{split}
& \alpha = \vec{w} \cdot (\vec{p} \times \vec{v}) \\
& \beta = \vec{w} \cdot (\vec{u} \times \vec{p})
\end{split}
\end{equation}
α = w ⋅ ( p × v ) β = w ⋅ ( u × p )
これであとはα , β \alpha,\beta α , β w ⃗ \vec{w} w Vec3 n = Cross ( m_u , m_v );
m_normal = n . normalized ();
m_d = Dot ( m_normal , m_q );
m_w = n / Dot ( n , n );
α , β \alpha,\beta α , β Vec3 p = ray . point_at ( t );
Vec3 planarHitPointVector = p - m_q ;
double alpha = Dot ( m_w , Cross ( planarHitPointVector , m_v ));
double beta = Dot ( m_w , Cross ( m_u , planarHitPointVector ));
// 範囲内？
Interval unit { 0.0 , 1.0 };
if ( ! unit . Contains ( alpha ) || ! unit . Contains ( beta )) { return false ; }
