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h = ± x 2 ( m 1 2 m 2 2 + m 1 2 ) ∓ x 1 ( m 1 2 m 2 2 + m 2 2 ) ∓ ( m 1 2 m 2 2 + m 1 2 ) ± ( m 1 2 m 2 2 + m 2 2 ) {\displaystyle h={\dfrac {\pm x_{2}\left({\sqrt {m_{1}{}^{2}m_{2}{}^{2}+m_{1}{}^{2}}}\right)\mp x_{1}\left({\sqrt {m_{1}{}^{2}m_{2}{}^{2}+m_{2}{}^{2}}}\right)}{\mp \left({\sqrt {m_{1}{}^{2}m_{2}{}^{2}+m_{1}{}^{2}}}\right)\pm \left({\sqrt {m_{1}{}^{2}m_{2}{}^{2}+m_{2}{}^{2}}}\right)}}\,}
For this one, choose either x1 and m1 or x2 and m2 to substitute in for x and m and of course the positive result is what you want:
r = ± ( x − h ) m 2 + 1 m {\displaystyle r=\pm {\dfrac {(x-h){\sqrt {m^{2}+1}}}{m}}\,}
From this, knowing 2 points and h and r, deriving k should be easy.
y = A x 3 + B x 2 + C x + D {\displaystyle y=Ax^{3}+Bx^{2}+Cx+D\,}
m = 3 A x 2 + 2 B x + C {\displaystyle m=3Ax^{2}+2Bx+C\,}
( x 1 , y 1 ) , m 1 ; ( x 2 , y 2 ) , m 2 {\displaystyle (x_{1},y_{1}),m_{1};(x_{2},y_{2}),m_{2}\,}
x 1 ≠ x 2 {\displaystyle x1\neq x2\,}
( m 1 = 3 A x 1 2 + 2 B x 1 + C ) − ( m 2 = 3 A x 2 2 + 2 B x 2 + C ) {\displaystyle {\begin{aligned}(m_{1}&=3Ax_{1}^{2}+2Bx_{1}+C)\\-(m_{2}&=3Ax_{2}^{2}+2Bx_{2}+C)\\\end{aligned}}}
m 1 − m 2 = 3 A ( x 1 2 − x 2 2 ) + 2 B ( x 1 − x 2 ) {\displaystyle m_{1}-m_{2}=3A(x_{1}^{2}-x_{2}^{2})+2B(x_{1}-x_{2})\,}
2 B ( x 1 − x 2 ) = m 1 − m 2 − 3 A ( x 1 2 − x 2 2 ) {\displaystyle 2B(x_{1}-x_{2})=m_{1}-m_{2}-3A(x_{1}^{2}-x_{2}^{2})}
B = m 1 − m 2 − 3 A ( x 1 2 − x 2 2 ) 2 x 1 − 2 x 2 {\displaystyle B={\frac {m_{1}-m_{2}-3A(x_{1}^{2}-x_{2}^{2})}{2x_{1}-2x_{2}}}}
2 B x = m − 3 A x 2 − C {\displaystyle 2Bx=m-3Ax^{2}-C\,}
2 B = m − 3 A x 2 − C x {\displaystyle 2B={\frac {m-3Ax^{2}-C}{x}}\,}
2 B = m 1 − 3 A x 1 2 − C x 1 − 2 B = m 2 − 3 A x 2 2 − C x 2 {\displaystyle {\begin{aligned}2B&={\frac {m_{1}-3Ax_{1}^{2}-C}{x_{1}}}\\-\quad 2B&={\frac {m_{2}-3Ax_{2}^{2}-C}{x_{2}}}\\\end{aligned}}}
0 = m 1 − 3 A x 1 2 − C x 1 − m 2 − 3 A x 2 2 − C x 2 {\displaystyle 0={\frac {m_{1}-3Ax_{1}^{2}-C}{x_{1}}}-{\frac {m_{2}-3Ax_{2}^{2}-C}{x_{2}}}}
0 = x 2 ( m 1 − 3 A x 1 2 − C ) − x 1 ( m 2 − 3 A x 2 2 − C ) {\displaystyle 0=x_{2}(m_{1}-3Ax_{1}^{2}-C)-x_{1}(m_{2}-3Ax_{2}^{2}-C)}
0 = m 1 x 2 − 3 A x 1 2 x 2 − C x 2 − m 2 x 1 + 3 A x 1 x 2 2 + C x 1 {\displaystyle 0=m_{1}x_{2}-3Ax_{1}^{2}x_{2}-Cx_{2}-m_{2}x_{1}+3Ax_{1}x_{2}^{2}+Cx_{1}}
C x 1 − C x 2 = m 2 x 1 − 3 A x 1 x 2 2 − m 1 x 2 + 3 A x 1 2 x 2 {\displaystyle Cx_{1}-Cx_{2}=m_{2}x_{1}-3Ax_{1}x_{2}^{2}-m_{1}x_{2}+3Ax_{1}^{2}x_{2}}
C ( x 1 − x 2 ) = m 2 x 1 − m 1 x 2 + 3 A x 1 x 2 ( x 1 − x 2 ) {\displaystyle C(x_{1}-x_{2})=m_{2}x_{1}-m_{1}x_{2}+3Ax_{1}x_{2}(x_{1}-x_{2})\,}
C = m 2 x 1 − m 1 x 2 + 3 A x 1 x 2 ( x 1 − x 2 ) x 1 − x 2 {\displaystyle C={\frac {m_{2}x_{1}-m_{1}x_{2}+3Ax_{1}x_{2}(x_{1}-x_{2})}{x_{1}-x_{2}}}\,}
y 1 = A x 1 3 + B x 1 2 + C x 1 + D − y 2 = A x 2 3 + B x 2 2 + C x 2 + D {\displaystyle {\begin{aligned}y_{1}&=Ax_{1}^{3}+Bx_{1}^{2}+Cx_{1}+D\\-\quad y_{2}&=Ax_{2}^{3}+Bx_{2}^{2}+Cx_{2}+D\\\end{aligned}}}
y 1 − y 2 = A ( x 1 3 − x 2 3 ) + B ( x 1 2 − x 2 2 ) + C ( x 1 − x 2 ) {\displaystyle y_{1}-y_{2}=A(x_{1}^{3}-x_{2}^{3})+B(x_{1}^{2}-x_{2}^{2})+C(x_{1}-x_{2})}
y 1 − y 2 = A ( x 1 3 − x 2 3 ) + ( m 1 − m 2 − 3 A ( x 1 2 − x 2 2 ) 2 x 1 − 2 x 2 ) ( x 1 2 − x 2 2 ) + ( m 2 x 1 − m 1 x 2 + 3 A x 1 x 2 ( x 1 − x 2 ) x 1 − x 2 ) ( x 1 − x 2 ) {\displaystyle y_{1}-y_{2}=A(x_{1}^{3}-x_{2}^{3})+\left({\frac {m_{1}-m_{2}-3A(x_{1}^{2}-x_{2}^{2})}{2x_{1}-2x_{2}}}\right)(x_{1}^{2}-x_{2}^{2})+\left({\frac {m_{2}x_{1}-m_{1}x_{2}+3Ax_{1}x_{2}(x_{1}-x_{2})}{x_{1}-x_{2}}}\right)(x_{1}-x_{2})}
y 1 − y 2 = A ( x 1 3 − x 2 3 ) + 1 2 ( x 1 + x 2 ) [ m 1 − m 2 − 3 A ( x 1 2 − x 2 2 ) ] + m 2 x 1 − m 1 x 2 + 3 A x 1 x 2 ( x 1 − x 2 ) {\displaystyle y_{1}-y_{2}=A(x_{1}^{3}-x_{2}^{3})+{1 \over 2}(x_{1}+x_{2})[m_{1}-m_{2}-3A(x_{1}^{2}-x_{2}^{2})]+m_{2}x_{1}-m_{1}x_{2}+3Ax_{1}x_{2}(x_{1}-x_{2})}
y 1 − y 2 = A ( x 1 3 − x 2 3 ) + 1 2 ( x 1 + x 2 ) ( m 1 − m 2 ) − 3 2 A ( x 1 + x 2 ) ( x 1 2 − x 2 2 ) + m 2 x 1 − m 1 x 2 + 3 A x 1 x 2 ( x 1 − x 2 ) {\displaystyle y_{1}-y_{2}=A(x_{1}^{3}-x_{2}^{3})+{1 \over 2}(x_{1}+x_{2})(m_{1}-m_{2})-{3 \over 2}A(x_{1}+x_{2})(x_{1}^{2}-x_{2}^{2})+m_{2}x_{1}-m_{1}x_{2}+3Ax_{1}x_{2}(x_{1}-x_{2})}
A ( x 1 3 − x 2 3 ) − 3 2 A ( x 1 + x 2 ) ( x 1 2 − x 2 2 ) + 3 A x 1 x 2 ( x 1 − x 2 ) = y 1 − y 2 − 1 2 ( x 1 + x 2 ) ( m 1 − m 2 ) + m 1 x 2 − m 2 x 1 {\displaystyle A(x_{1}^{3}-x_{2}^{3})-{3 \over 2}A(x_{1}+x_{2})(x_{1}^{2}-x_{2}^{2})+3Ax_{1}x_{2}(x_{1}-x_{2})=y_{1}-y_{2}-{1 \over 2}(x_{1}+x_{2})(m_{1}-m_{2})+m_{1}x_{2}-m_{2}x_{1}}
A [ ( x 1 3 − x 2 3 ) − 3 2 ( x 1 + x 2 ) ( x 1 2 − x 2 2 ) + 3 x 1 x 2 ( x 1 − x 2 ) ] = y 1 − y 2 − 1 2 ( x 1 + x 2 ) ( m 1 − m 2 ) + m 1 x 2 − m 2 x 1 {\displaystyle A[(x_{1}^{3}-x_{2}^{3})-{3 \over 2}(x_{1}+x_{2})(x_{1}^{2}-x_{2}^{2})+3x_{1}x_{2}(x_{1}-x_{2})]=y_{1}-y_{2}-{1 \over 2}(x_{1}+x_{2})(m_{1}-m_{2})+m_{1}x_{2}-m_{2}x_{1}}
A [ x 1 3 − x 2 3 − 3 2 x 1 3 + 3 2 x 1 x 2 2 − 3 2 x 1 2 x 2 + 3 2 x 2 3 + 3 x 1 2 x 2 − 3 x 1 x 2 2 ] {\displaystyle A[x_{1}^{3}-x_{2}^{3}-{3 \over 2}x_{1}^{3}+{3 \over 2}x_{1}x_{2}^{2}-{3 \over 2}x_{1}^{2}x_{2}+{3 \over 2}x_{2}^{3}+3x_{1}^{2}x_{2}-3x_{1}x_{2}^{2}]} = y 1 − y 2 − 1 2 m 1 x 1 + 1 2 m 2 x 1 − 1 2 m 1 x 2 + 1 2 m 2 x 2 + m 1 x 2 − m 2 x 1 {\displaystyle =y_{1}-y_{2}-{1 \over 2}m_{1}x_{1}+{1 \over 2}m_{2}x_{1}-{1 \over 2}m_{1}x_{2}+{1 \over 2}m_{2}x_{2}+m_{1}x_{2}-m_{2}x_{1}}
A [ 1 2 x 2 3 − 3 2 x 1 x 2 2 + 3 2 x 1 2 x 2 − 1 2 x 1 3 ] {\displaystyle A[{1 \over 2}x_{2}^{3}-{3 \over 2}x_{1}x_{2}^{2}+{3 \over 2}x_{1}^{2}x_{2}-{1 \over 2}x_{1}^{3}]} = y 1 − y 2 − 1 2 m 1 x 1 − 1 2 m 2 x 1 + 1 2 m 1 x 2 + 1 2 m 2 x 2 {\displaystyle =y_{1}-y_{2}-{1 \over 2}m_{1}x_{1}-{1 \over 2}m_{2}x_{1}+{1 \over 2}m_{1}x_{2}+{1 \over 2}m_{2}x_{2}}
A [ x 2 3 − 3 x 1 x 2 2 + 3 x 1 2 x 2 − x 1 3 ] {\displaystyle A[x_{2}^{3}-3x_{1}x_{2}^{2}+3x_{1}^{2}x_{2}-x_{1}^{3}]} = 2 y 1 − 2 y 2 − m 1 x 1 − m 2 x 1 + m 1 x 2 + m 2 x 2 {\displaystyle =2y_{1}-2y_{2}-m_{1}x_{1}-m_{2}x_{1}+m_{1}x_{2}+m_{2}x_{2}\,}
A ( x 2 − x 1 ) 3 = 2 y 1 − 2 y 2 − ( m 1 + m 2 ) ( x 1 − x 2 ) {\displaystyle A(x_{2}-x_{1})^{3}=2y_{1}-2y_{2}-(m_{1}+m_{2})(x_{1}-x_{2})\,}
A = 2 y 1 − 2 y 2 − ( m 1 + m 2 ) ( x 1 − x 2 ) ( x 2 − x 1 ) 3 {\displaystyle A={\frac {2y_{1}-2y_{2}-(m_{1}+m_{2})(x_{1}-x_{2})}{(x_{2}-x_{1})^{3}}}}
λ p l a n c k = [ ∂ ( σ T e f f 4 ) ∂ T s ] − 1 {\displaystyle \lambda _{planck}=\left[{\frac {\partial \left(\sigma T_{eff}^{4}\right)}{\partial T_{s}}}\right]^{-1}}
![{\displaystyle y_{1}-y_{2}=A(x_{1}^{3}-x_{2}^{3})+{1 \over 2}(x_{1}+x_{2})[m_{1}-m_{2}-3A(x_{1}^{2}-x_{2}^{2})]+m_{2}x_{1}-m_{1}x_{2}+3Ax_{1}x_{2}(x_{1}-x_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a846c3b2a725afd55952152733d960f8b93d4d3)
![{\displaystyle A[(x_{1}^{3}-x_{2}^{3})-{3 \over 2}(x_{1}+x_{2})(x_{1}^{2}-x_{2}^{2})+3x_{1}x_{2}(x_{1}-x_{2})]=y_{1}-y_{2}-{1 \over 2}(x_{1}+x_{2})(m_{1}-m_{2})+m_{1}x_{2}-m_{2}x_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/685ffe1b8b730d6367c639c611061370d16fb864)
![{\displaystyle A[x_{1}^{3}-x_{2}^{3}-{3 \over 2}x_{1}^{3}+{3 \over 2}x_{1}x_{2}^{2}-{3 \over 2}x_{1}^{2}x_{2}+{3 \over 2}x_{2}^{3}+3x_{1}^{2}x_{2}-3x_{1}x_{2}^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e73506ceb0576233e12c5c12ca1b17f3fe158f6)
![{\displaystyle A[{1 \over 2}x_{2}^{3}-{3 \over 2}x_{1}x_{2}^{2}+{3 \over 2}x_{1}^{2}x_{2}-{1 \over 2}x_{1}^{3}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91dacb35280e12b60e32aca61bf69a8ab33bf4a6)
![{\displaystyle A[x_{2}^{3}-3x_{1}x_{2}^{2}+3x_{1}^{2}x_{2}-x_{1}^{3}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/448e190f2250c711e96afeaa6a838aa4aa490c82)
![{\displaystyle \lambda _{planck}=\left[{\frac {\partial \left(\sigma T_{eff}^{4}\right)}{\partial T_{s}}}\right]^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2484b5f9574d944ed2e0c5b1522bce3ed27155ac)
