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I'm wondering how to get all points (x,y) that constitute a given parabola. It's barely to simulate a jump of an object from point A to point B. I know: -The coordinate of point A -The coordinate of point B -The coordinate of the parabola's top

What i'm trying to do is to loop from start_x to dest_x to calculate the Y-coordinate at that moment, but I don't know how to do that part.

pixels = []
for x in start_x to dest_x
   y = (calculate the Y-coord here based on x )
   pixels.push( [x,y] )
end
return pixels

My math skills are not exceptional. Thank you.

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  • \$\begingroup\$ you'll need a third point to specify the parabola, or the direction of the "gravity" and its strength. \$\endgroup\$
    – ratchet freak
    Oct 15, 2015 at 21:43
  • \$\begingroup\$ You're using too much data. If you know the parabola's top coordinate then you'll need one more point only and everything is done, but you have a second point too and this can lead you to undesired results \$\endgroup\$
    – liggiorgio
    Oct 16, 2015 at 0:14

1 Answer 1

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In general, there is no parabola that goes through three given points. However, if you loosen the problem constraints and only define the Y coordinate of the parabola’s top and allow its X coordinate to be free, there is usually a solution.

Given A.x, A.y, B.x, B.y (A and B’s coordinates) and C.y (the parabola’s top), one can find a parabola formula of the form:

f(x) = C.y - u (x - v)²

One needs to find u and v in this formula. The constraints give us, after some maths:

k = sqrt((C.y - A.y) / (C.y - B.y))
v = (k B.x - A.x) / (k + 1)
u = (C.y - A.y) / (A.x - v)²

For instance if A = (0, 0), B = (10, 5) and C.y = 15:

k = sqrt((15 - 0) / (15 - 5)) = 1.2247448714
v = (1.2247448714 * 10 - 0) / (1.2247448714 + 1) = 5.505102572
u = (15 - 0) / (0 - 5.505102572)² = 0.494948974

Hence the formula:

f(x) = 30 - 0.494948974 * (x - 5.505102572)²

You can see this parabola going from (0,0) to (10,5) with a peak at Y = 15, as expected:

plot of the example parabloa

The following code should work:

pixels = []
k = sqrt((top_y - start_y) / (top_y - dest_y))
v = (k * dest_x - start_x) / (k + 1)
u = (top_y - start_y) / ((start_x - v) * (start_x - v))
for x in start_x to dest_x
   y = top_y - u * (x - v) * (x - v)
   pixels.push( [x,y] )
end
return pixels
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  • \$\begingroup\$ Many thanks sam hocevar ! I'll let you know if it's applicable. \$\endgroup\$
    – Wolfrevo stack
    Oct 16, 2015 at 14:11
  • \$\begingroup\$ Unfortunately, i get some weird results. When the top_y is same as start_y, the movement becomes straight. It's normal. But when top_y is different, the movement become weirds. I don't know how to explain that. But after some investigations, il appears that the y at the beginning of the move is too inappropriate. It starts at 5838, for example. Whereas my start_y = ~200 and same for dest_y. \$\endgroup\$
    – Wolfrevo stack
    Oct 17, 2015 at 11:39
  • \$\begingroup\$ Please be more specific. Give the values for Ax, Ay, Bx, By, C.y, and what you get for k, u and v. Otherwise it’ll be difficult to tell what’s wrong. \$\endgroup\$
    – sam hocevar
    Oct 17, 2015 at 13:58
  • \$\begingroup\$ Ha sorry. Ax = 120; Ay = 258; Bx = 378; By = 264; Cy = 228; k = 0.92582009...; u = -85.852319..; v = 119,408867.... \$\endgroup\$
    – Wolfrevo stack
    Oct 17, 2015 at 16:22
  • \$\begingroup\$ Ah, but then Cy is smaller than Ay and By, so it’s no longer a “jump” above an object. If your Y axis goes downwards, you’ll need to use k = -sqrt(…) instead of k = sqrt(…). Also, check your code, because with the formulas above you should get k=0.912870929175277. \$\endgroup\$
    – sam hocevar
    Oct 18, 2015 at 0:28

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