# Point on a line closest to x,y

I asked this over on stackoverflow and it transpired it was amenable to a mathematical solution. So I am seeking clarity here, or I go night night.

//Returns the point on the line traced from start to end which
//comes nearest to 0.5, 0.5.
fpPoint closestToCentre(fpPoint start, fpPoint end);


Anyone know of quicker way than single stepping through the pixels?

Now I know how to rotate the gradient $90^o$ and can find which way easy enough, so it is now a matter of finding a line intercept and expressing this mathematically so I can solve for $x$ and $y$.

I was thinking my original line is known and : $dy_1$ = $dx_1\left(\frac{dy}{dx_1}\right) + c_1$

and the perpendicular through center is : $dy_2 = dx_2 \left( \frac{dy}{dx_2}\right) + c_2$

I don't really need trig (and to worry about negativity in certain quadrants) to rotate the grad $90^o$ do I? Night all.

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You want the point $(x,y)$ to be aligned with the points $(x_s,y_s)$ and $(x_e,y_e)$ and the line they make to be perpendicular to the line $(x,y)$ and $(x_o,y_o)$ make, where $x_o=y_o=.5$.

The first condition reads $$(x-x_s)(y_e-y_s)-(y-y_s)(x_e-x_s)=0.$$ The second condition reads $$(x-x_o)(x_e-x_s)+(y-y_o)(y_e-y_s)=0.$$ This is a Cramer system for the unknown $(x,y)$ hence $x$ and $y$ are ratios of $2\times2$ determinants depending on the parameters $(x_s,y_s,x_e,y_e,x_o,y_o)$.

Note that the common denominator of the expressions of $x$ and $y$ is $$D=(x_e-x_s)^2+(y_e-y_s)^2,$$ hence $D\ne0$ unless the points $(x_s,y_s)$ and $(x_e,y_e)$ coincide, in which case they do not define a line.

A second method is to look for $(x,y)$ as $(x,y)=(x_s,y_s)+t\cdot(x_e-x_s,y_e-y_s)$ for a given scalar $t$, and to plug this into the second condition. This yields $$D\cdot t=(x_o-x_s)(x_e-x_s)+(y_o-y_s)(y_e-y_s),$$ hence $$x=x_s+\frac{(x_o-x_s)(x_e-x_s)+(y_o-y_s)(y_e-y_s)}{(x_e-x_s)^2+(y_e-y_s)^2}\cdot(x_e-x_s),$$ and $$y=y_s+\frac{(x_o-x_s)(x_e-x_s)+(y_o-y_s)(y_e-y_s)}{(x_e-x_s)^2+(y_e-y_s)^2}\cdot(y_e-y_s).$$

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Both work, but I like the parametric approach more... :) – J. M. Aug 22 '11 at 3:03
I like it, the second I can almost understand. For real thickies like me I needed graphics which I was denied as a newb here. topcoder.com/tc?d1=tutorials&d2=geometry1&module=Static – John Aug 22 '11 at 9:27