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How can I simulate the gravity of a sphere instead of a plane?

For a naval warfare game, I wish to implement a surface that simulates a curved plane with horizon. The plane is rendered with water physics, which needs, along with all physical objects, avatars and projectiles, to respond to gravity as a focal point rather than a distributed surface.

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    \$\begingroup\$ To answer this question we would need to know how you are simulating gravity right now and why this system can not be easily generalized to a vector facing a point instead of one facing straight down. \$\endgroup\$ – Philipp May 11 '15 at 14:24
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    \$\begingroup\$ Also, how familiar are you with vector math in general? \$\endgroup\$ – Philipp May 11 '15 at 14:26
  • \$\begingroup\$ The title says "sphere", but the body says "a curved plane". Which do you mean? \$\endgroup\$ – Anko May 16 '15 at 8:52
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It depends on the effect you are looking for.

If you are looking for a constant gravity the answer is:

object.AddForce(((Vector) sphere.center - (Vector) object.pos).normalized * gravityForce);

If you are looking for a gravity based on the distance between the object and the sphere then you replace gravityForce with this formula:

gravityForce =  6.673×10^(−11) * sphere.mass * object.mass / Distance(sphere.center, object.pos); 
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    \$\begingroup\$ If you go with the second route, masses must be in kilograms. \$\endgroup\$ – El'endia Starman May 11 '15 at 20:27
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    \$\begingroup\$ @El'endiaStarman It's worth noting that the gravitational constant (6.673×10^(−11) in the above) can be adjusted for other units of mass. \$\endgroup\$ – chbaker0 May 11 '15 at 20:50
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You simply need a reworked Newton's law of universal gravitation.

So F=G(M1*M2)/R2 translates as code into.

// Create force scalar based off of distance.

distanceSquared = (gravityCenter.y - object.y)^2 + (gravityCenter.x - object.x)^2

force = gravitationalConstant*distanceSquared

// Get direction from object to gravity epicenter.

direction = arctangent2(gravityCenter.y - object.y, gravityCenter.x - object.x)

// Transform vector into into coordinate based impulse.

object.xVelocity += force*cosine(direction)
object.yVelocity += force*sine(direction)

Then adjust gravitationalConstant to suit.

NOTE: Unless you want to send ships into space, implementing gravity based on distance is completely a waste of cycles as a constant "downwards" force is more processor friendly.

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