Mixing
Alternative Parabola Formula for Fortune's Algorithm using Strange Subscripts $f$ and $d$.
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I am trying to figure out how to create a voronoi diagram using Fortune's algorithm. I have found a pretty good explanation of the algorithm here. However, I don't understand his style of math notation. I haven't seen subscripts used in mathematics before except in discrete math, and I am curious what these represent.
What do the subscript $f$ and $d$ represent here?
Sample from linked website:
I mentioned above that the algorithm represents growing cells as parabolas. In school most people are taught that a parabola is defined by the equation $y = ax^2 + bx + c$. However there is another definition: Take a point (which we will call the focus) and a straight line (which we will call the directrix). Now if we were to mark all of the points on the plane for which the distance to the focus is the same as the distance to the closest point on the directrix, the resulting set of points would make a parabola.
With some manipulation of the standard definition of the parabola and our definition of the focus and directrix, we can show that for a directrix $y = y_d$ and focus $(x_f, y_f)$, we get the formula $y = frac{1}{2(y_f, y_d)}(x - x_f)^2 + frac{y_f + y_d}{2}$
algorithms computational-mathematics computational-geometry
asked Jan 29 at 3:30
LuminousNutriaLuminousNutria
46912
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add a comment |
$begingroup$
I am trying to figure out how to create a voronoi diagram using Fortune's algorithm. I have found a pretty good explanation of the algorithm here. However, I don't understand his style of math notation. I haven't seen subscripts used in mathematics before except in discrete math, and I am curious what these represent.
What do the subscript $f$ and $d$ represent here?
Sample from linked website:
I mentioned above that the algorithm represents growing cells as parabolas. In school most people are taught that a parabola is defined by the equation $y = ax^2 + bx + c$. However there is another definition: Take a point (which we will call the focus) and a straight line (which we will call the directrix). Now if we were to mark all of the points on the plane for which the distance to the focus is the same as the distance to the closest point on the directrix, the resulting set of points would make a parabola.
With some manipulation of the standard definition of the parabola and our definition of the focus and directrix, we can show that for a directrix $y = y_d$ and focus $(x_f, y_f)$, we get the formula $y = frac{1}{2(y_f, y_d)}(x - x_f)^2 + frac{y_f + y_d}{2}$
algorithms computational-mathematics computational-geometry
asked Jan 29 at 3:30
LuminousNutriaLuminousNutria
46912
$endgroup$
$begingroup$
d for directrix and f for focus?
$endgroup$
– VorKir
Feb 3 at 6:16
add a comment |
$begingroup$
I am trying to figure out how to create a voronoi diagram using Fortune's algorithm. I have found a pretty good explanation of the algorithm here. However, I don't understand his style of math notation. I haven't seen subscripts used in mathematics before except in discrete math, and I am curious what these represent.
What do the subscript $f$ and $d$ represent here?
Sample from linked website:
I mentioned above that the algorithm represents growing cells as parabolas. In school most people are taught that a parabola is defined by the equation $y = ax^2 + bx + c$. However there is another definition: Take a point (which we will call the focus) and a straight line (which we will call the directrix). Now if we were to mark all of the points on the plane for which the distance to the focus is the same as the distance to the closest point on the directrix, the resulting set of points would make a parabola.
With some manipulation of the standard definition of the parabola and our definition of the focus and directrix, we can show that for a directrix $y = y_d$ and focus $(x_f, y_f)$, we get the formula $y = frac{1}{2(y_f, y_d)}(x - x_f)^2 + frac{y_f + y_d}{2}$
algorithms computational-mathematics computational-geometry
asked Jan 29 at 3:30
LuminousNutriaLuminousNutria
46912
$endgroup$
I am trying to figure out how to create a voronoi diagram using Fortune's algorithm. I have found a pretty good explanation of the algorithm here. However, I don't understand his style of math notation. I haven't seen subscripts used in mathematics before except in discrete math, and I am curious what these represent.
What do the subscript $f$ and $d$ represent here?
Sample from linked website:
I mentioned above that the algorithm represents growing cells as parabolas. In school most people are taught that a parabola is defined by the equation $y = ax^2 + bx + c$. However there is another definition: Take a point (which we will call the focus) and a straight line (which we will call the directrix). Now if we were to mark all of the points on the plane for which the distance to the focus is the same as the distance to the closest point on the directrix, the resulting set of points would make a parabola.
With some manipulation of the standard definition of the parabola and our definition of the focus and directrix, we can show that for a directrix $y = y_d$ and focus $(x_f, y_f)$, we get the formula $y = frac{1}{2(y_f, y_d)}(x - x_f)^2 + frac{y_f + y_d}{2}$
algorithms computational-mathematics computational-geometry
algorithms computational-mathematics computational-geometry
asked Jan 29 at 3:30
LuminousNutriaLuminousNutria
46912
asked Jan 29 at 3:30
LuminousNutriaLuminousNutria
46912
edited Jan 29 at 3:35
LuminousNutria
asked Jan 29 at 3:30
LuminousNutriaLuminousNutria
46912
asked Jan 29 at 3:30
LuminousNutriaLuminousNutria
46912
asked Jan 29 at 3:30
LuminousNutriaLuminousNutria
46912
46912
$begingroup$
d for directrix and f for focus?
$endgroup$
– VorKir
Feb 3 at 6:16
add a comment |
$begingroup$
d for directrix and f for focus?
$endgroup$
– VorKir
Feb 3 at 6:16
$begingroup$
d for directrix and f for focus?
$endgroup$
– VorKir
Feb 3 at 6:16
$begingroup$
d for directrix and f for focus?
$endgroup$
– VorKir
Feb 3 at 6:16
add a comment |
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$begingroup$
d for directrix and f for focus?
$endgroup$
– VorKir
Feb 3 at 6:16