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 Post subject: Firing Arrows in a Cartesian Plane
PostPosted: Sun Jan 26, 2014 4:05 pm 
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Before I pose the question i have, let me give a bit of backstory:

I've been working on a video game project recently, and one of the bits of code I had to write was to set initial movement vectors to an object so it would follow a ballistic trajectory and go through a specified point. There were three circumstances I had to code for: a constant maximum height so the direction and magnitude of the vector had to be determined, a constant direction so the magnitude of the vector had to be determined, and a constant magnitude so the direction had to be determined. The first two cases were not very interesting, but the last case brought about an interesting problem. There are areas of the grid that a fired projectile obviously cannot hit, and areas that it can. What I was interested in was where those areas actually are.

If a projectile is fired along a ballistic trajectory with a predetermined speed and gravity on a cartesian graph, what is the equation of the line that defines where the projectile cannot possibly reach?

I think it's a conic section, although I have no way to be sure. Does anyone know how to solve this?

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 Post subject: Re: Firing Arrows in a Cartesian Plane
PostPosted: Thu Feb 06, 2014 2:15 am 
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It's a parabola, called the parabola of safety. The French wiki page has more explanation, including the final equation y = h - x^2/4h.
This gives y(2h)=0, showing that a projectile can reach twice as far horizontally than it can reach when fired straight up.

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 Post subject: Re: Firing Arrows in a Cartesian Plane
PostPosted: Thu Feb 06, 2014 12:53 pm 
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Thank you for the link, Jaap. That article was very helpful despite the fact that I don't speak French. But now I have another (slightly more relevant) question: What are the equation(s) of the parabola(s) that pass through an arbitrary point within the "parabola of safety?" I think I can figure out how to determine the initial angle from that equation.

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 Post subject: Re: Firing Arrows in a Cartesian Plane
PostPosted: Thu Feb 06, 2014 1:14 pm 
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Jorbs3210 wrote:
Thank you for the link, Jaap. That article was very helpful despite the fact that I don't speak French. But now I have another (slightly more relevant) question: What are the equation(s) of the parabola(s) that pass through an arbitrary point within the "parabola of safety?" I think I can figure out how to determine the initial angle from that equation.

Check out the graph that shows the line OC passing through the focus of the parabola. It should be easy to compute the parabola by first computing the location of the focus (the point the horizontally fired object (blue curve) crosses the OC line) and then from the focus and points O and C computing the resulting parabola.

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