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Here's part of a solution. I didn't get to finish it in time. I'll try again later.

If I understand correctly, you have a planet's position & velocity, as well as a ship's position and speed. You want to get the ship's movement direction. I'm assuming the ship's and planet's speeds are constant. I also assume, without loss of generality, that the ship is at (0,0); to do this, subtract the ship's position from the planet's, and add the ship's position back onto the result of the operation described below.

Unfortunately, without latex, I can't format this answer very well, but we'll attempt to make do. Let:

• s_s = the ship's speed (s_s.x, s_s.y, likewise)
• s_a = the ship's bearing (angle of movement, what we want to calculate)
• p_p = the planet's initial position, global coords
• p_r = the planet's distance (radius) from the center of orbit, derivable from p_p
• p_a = the planet's initial angle in radians, relative to the center of orbit
• p_s = the planet's angular velocity (rad/sec)
• t = the time to collision (this turns out to be something we must calculate as well)

Here's the equations for the position of the two bodies, broken down into components:

ship.x = s_s.x * t * cos(s_a)
ship.y = s_s.y * t * sin(s_a)

planet.x = p_r * cos(p_a + p_s * t) + p_p.x
planet.y = p_r * sin(p_a + p_s * t) + p_p.y

Since we want ship.x = planet.x and ship.y = planet.y at some instant t, we obtain this equation (the y case is nearly symmetrical):

   s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x
   s_s.y * t * sin(s_a) = p_r * sin(p_a + p_s * t) + p_p.y

Solving the top equation for s_a:

   s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x
=> s_a = arccos((p_r * cos(p_a + p_s * t) + p_p.x) / (s_s.x * t))

Substituting this into the second equation results in a fairly terrifying equation that Wolfram alpha won't solve for me. There may be a better way to do this not involving polar coordinates. If anyone wants to give this method a shot, you're welcome to it; I've made this a wiki. Otherwise, you may want to take this to the Math StackExchange.

Here's part of a solution. I didn't get to finish it in time. I'll try again later.

If I understand correctly, you have a planet's position & velocity, as well as a ship's position and speed. You want to get the ship's movement direction. I'm assuming the ship's and planet's speeds are constant. I also assume, without loss of generality, that the ship is at \$(0,0)\$; to do this, subtract the ship's position from the planet's, and add the ship's position back onto the result of the operation described below.

Let:

• \$s_s\$ = the ship's speed (\$s_s.x, s_s.y\$, likewise)
• \$s_a\$ = the ship's bearing (angle of movement, what we want to calculate)
• \$p_p\$ = the planet's initial position, global coords
• \$p_r\$ = the planet's distance (radius) from the center of orbit, derivable from \$p_p\$
• \$p_a\$ = the planet's initial angle in radians, relative to the center of orbit
• \$p_s\$ = the planet's angular velocity (rad/sec)
• \$t\$ = the time to collision (this turns out to be something we must calculate as well)

Here's the equations for the position of the two bodies, broken down into components: $$ ship.x = s_s.x * t * cos(s_a) $$ $$ ship.y = s_s.y * t * sin(s_a) $$ $$ planet.x = p_r * cos(p_a + p_s * t) + p_p.x $$ $$ planet.y = p_r * sin(p_a + p_s * t) + p_p.y $$

Since we want \$ship.x = planet.x\$ and \$ship.y = planet.y\$ at some instant \$t\$, we obtain this equation (the \$y\$ case is nearly symmetrical):

$$ s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x $$ $$ s_s.y * t * sin(s_a) = p_r * sin(p_a + p_s * t) + p_p.y $$

Solving the top equation for \$s_a\$:

$$ s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x \\ \implies s_a = \frac{arccos((p_r * cos(p_a + p_s * t) + p_p.x)}{(s_s.x * t))}$$

Substituting this into the second equation results in a fairly terrifying equation that Wolfram alpha won't solve for me. There may be a better way to do this not involving polar coordinates. If anyone wants to give this method a shot, you're welcome to it; I've made this a wiki. Otherwise, you may want to take this to the Math StackExchange.

Here's part of a solution. I didn't get to finish it in time. I'll try again later.

If I understand correctly, you have a planet's position & velocity, as well as a ship's position and speed. You want to get the ship's movement direction. I'm assuming the ship's and planet's speeds are constant. I also assume, without loss of generality, that the ship is at \$(0,0)\$; to do this, subtract the ship's position from the planet's, and add the ship's position back onto the result of the operation described below.

Let:

• \$s_s\$ = the ship's speed (\$s_s.x, s_s.y\$, likewise)
• \$s_a\$ = the ship's bearing (angle of movement, what we want to calculate)
• \$p_p\$ = the planet's initial position, global coords
• \$p_r\$ = the planet's distance (radius) from the center of orbit, derivable from \$p_p\$
• \$p_a\$ = the planet's initial angle in radians, relative to the center of orbit
• \$p_s\$ = the planet's angular velocity (rad/sec)
• \$t\$ = the time to collision (this turns out to be something we must calculate as well)

Here's the equations for the position of the two bodies, broken down into components: $$ ship.x = s_s.x * t * cos(s_a) $$ $$ ship.y = s_s.y * t * sin(s_a) $$ $$ planet.x = p_r * cos(p_a + p_s * t) + p_p.x $$ $$ planet.y = p_r * sin(p_a + p_s * t) + p_p.y $$

Since we want \$ship.x = planet.x\$ and \$ship.y = planet.y\$ at some instant \$t\$, we obtain this equation (the \$y\$ case is nearly symmetrical):

$$ s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x $$ $$ s_s.y * t * sin(s_a) = p_r * sin(p_a + p_s * t) + p_p.y $$

Solving the top equation for \$s_a\$:

$$ s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x \\ \implies s_a = \frac{arccos((p_r * cos(p_a + p_s * t) + p_p.x)}{(s_s.x * t))}$$

Substituting this into the second equation results in a fairly terrifying equation that Wolfram alpha won't solve for me. There may be a better way to do this not involving polar coordinates. If anyone wants to give this method a shot, you're welcome to it; I've made this a wiki. Otherwise, you may want to take this to the Math StackExchange.

Here's part of a solution. I didn't get to finish it in time. I'll try again later.

If I understand correctly, you have a planet's position & velocity, as well as a ship's position and speed. You want to get the ship's movement direction. I'm assuming the ship's and planet's speeds are constant. I also assume, without loss of generality, that the ship is at (0,0); to do this, subtract the ship's position from the planet's, and add the ship's position back onto the result of the operation described below.

Unfortunately, without latex, I can't format this answer very well, but we'll attempt to make do. Let:

• s_s = the ship's speed (s_s.x, s_s.y, likewise)
• s_a = the ship's bearing (angle of movement, what we want to calculate)
• p_p = the planet's initial position, global coords
• p_r = the planet's distance (radius) from the center of orbit, derivable from p_p
• p_a = the planet's initial angle in radians, relative to the center of orbit
• p_s = the planet's angular velocity (rad/sec)
• t = the time to collision (this turns out to be something we must calculate as well)

Here's the equations for the position of the two bodies, broken down into components:

ship.x = s_s.x * t * cos(s_a)
ship.y = s_s.y * t * sin(s_a)

planet.x = p_r * cos(p_a + p_s * t) + p_p.x
planet.y = p_r * sin(p_a + p_s * t) + p_p.y

Since we want ship.x = planet.x and ship.y = planet.y at some instant t, we obtain this equation (the y case is nearly symmetrical):

   s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x
   s_s.y * t * sin(s_a) = p_r * sin(p_a + p_s * t) + p_p.y

Solving the top equation for s_a:

   s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x
=> s_a = arccos((p_r * cos(p_a + p_s * t) + p_p.x) / (s_s.x * t))

Substituting this into the second equation results in a fairly terrifying equation that Wolfram alpha won't solve for me. There may be a better way to do this not involving polar coordinates. If anyone wants to give this method a shot, you're welcome to it; I've made this a wiki. Otherwise, you may want to take this to the Math StackExchange.

Here's part of a solution. I didn't get to finish it in time. I'll try again later.

If I understand correctly, you have a planet's position & velocity, as well as a ship's position and speed. You want to get the ship's movement direction. I'm assuming the ship's and planet's speeds are constant. I also assume, without loss of generality, that the ship is at (0,0); to do this, subtract the ship's position from the planet's, and add the ship's position back onto the result of the operation described below.

Unfortunately, without latex, I can't format this answer very well, but we'll attempt to make do. Let:

• s_s = the ship's speed (s_s.x, s_s.y, likewise)
• s_a = the ship's bearing (angle of movement, what we want to calculate)
• p_p = the planet's initial position, global coords
• p_r = the planet's distance (radius) from the center of orbit, derivable from p_p
• p_a = the planet's initial angle in radians, relative to the center of orbit
• p_s = the planet's angular velocity (rad/sec)
• t = the time to collision (this turns out to be something we must calculate as well)

Here's the equations for the position of the two bodies, broken down into components:

ship.x = s_s.x * t * cos(s_a)
ship.y = s_s.y * t * sin(s_a)

planet.x = p_r * cos(p_a + p_s * t) + p_p.x
planet.y = p_r * sin(p_a + p_s * t) + p_p.y

Since we want ship.x = planet.x and ship.y = planet.y at some instant t, we obtain this equation (the y case is nearly symmetrical):

   s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x
   s_s.y * t * sin(s_a) = p_r * sin(p_a + p_s * t) + p_p.y

Solving the top equation for s_a:

   s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x
=> s_a = arccos((p_r * cos(p_a + p_s * t) + p_p.x) / (s_s.x * t))

Substituting this into the second equation results in a fairly terrifying equation that Wolfram alpha won't solve for me. There may be a better way to do this not involving polar coordinates. If anyone wants to give this method a shot, you're welcome to it; I've made this a wiki. Otherwise, you may want to take this to the Math StackExchange.

Here's part of a solution. I didn't get to finish it in time. I'll try again later.

If I understand correctly, you have a planet's position & velocity, as well as a ship's position and speed. You want to get the ship's movement direction. I'm assuming the ship's and planet's speeds are constant. I also assume, without loss of generality, that the ship is at \$(0,0)\$; to do this, subtract the ship's position from the planet's, and add the ship's position back onto the result of the operation described below.

Let:

• \$s_s\$ = the ship's speed (\$s_s.x, s_s.y\$, likewise)
• \$s_a\$ = the ship's bearing (angle of movement, what we want to calculate)
• \$p_p\$ = the planet's initial position, global coords
• \$p_r\$ = the planet's distance (radius) from the center of orbit, derivable from \$p_p\$
• \$p_a\$ = the planet's initial angle in radians, relative to the center of orbit
• \$p_s\$ = the planet's angular velocity (rad/sec)
• \$t\$ = the time to collision (this turns out to be something we must calculate as well)

Here's the equations for the position of the two bodies, broken down into components: $$ ship.x = s_s.x * t * cos(s_a) $$ $$ ship.y = s_s.y * t * sin(s_a) $$ $$ planet.x = p_r * cos(p_a + p_s * t) + p_p.x $$ $$ planet.y = p_r * sin(p_a + p_s * t) + p_p.y $$

Since we want \$ship.x = planet.x\$ and \$ship.y = planet.y\$ at some instant \$t\$, we obtain this equation (the \$y\$ case is nearly symmetrical):

$$ s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x $$ $$ s_s.y * t * sin(s_a) = p_r * sin(p_a + p_s * t) + p_p.y $$

Solving the top equation for \$s_a\$:

$$ s_s.x * t * cos(s_a) = p_r * cos(p_a + p_s * t) + p_p.x \\ \implies s_a = \frac{arccos((p_r * cos(p_a + p_s * t) + p_p.x)}{(s_s.x * t))}$$

Substituting this into the second equation results in a fairly terrifying equation that Wolfram alpha won't solve for me. There may be a better way to do this not involving polar coordinates. If anyone wants to give this method a shot, you're welcome to it; I've made this a wiki. Otherwise, you may want to take this to the Math StackExchange.