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Game Maker 8.0’s particle system can be a bit confusing at first. There isn’t a lot of documentation or examples of how it needs to be built in order to work effectively. There are also a huge amount of functions available and it can be hard to know where to start.

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Suppose you have a plane equation in local space and you’d like to express that plane equation in world space. The plane in local space is written as: [ P:= (n, w) ] where (n ) is the plane normal and (w ) is the plane offset.

A point (x ) is on the plane if [n cdot x = w ] Now define a transform (A ) as [ A:= (R, p) ] where (R ) is an orthonormal rotation matrix and (p ) is a translation vector. Suppose we have a transform (A ) that transforms points in local space into world space. With our transform (A ) we can convert any point (x_1 ) in local space (space 1) into world space (space 2): [x_2 = R x_1 + p ] Also any vector (n_1 ) in local space can be converted to world space: [n_2 = R n_1 ] Also suppose we have a plane defined in local space (space 1). Then for any point (x_1 ) in local space: [n_1 cdot x_1 = w_1 ] The main problem now is to find (w_2 ), the plane offset in world space. We can achieve this by substitution. First invert the transform relations above: [x_1 = R^T (x_2 – p) ] [n_1 = R^T n_2 ] where (R^T ) is the transpose of (R ). Recall that the inverse of an orthonormal matrix is the equal to the transpose.

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Now substitute these expressions into the local space plane equation: [R^T n_2 cdot (R^T (x_2 – p)) = w_1 ] Expand: [R^T n_2 cdot R^T x_2 – R^T n_2 cdot R^T p = w_1 ] The rotations cancel out since they are orthonormal. Also the dot product is equivalent to matrix multiplication by the transpose. For example: [R^T n_2 cdot R^T x_2 = n_2^T R R^T x_2 = n_2^T I x_2 = n_2 cdot x_2 ] Simplify: [n_2 cdot x_2 = w_1 + n_2 cdot p ] From this we can identify the world space plane offset (w_2 ): [w_2 = w_1 + n_2 cdot p ] Done! Update: Of course it is easy to transform a plane if you have it expressed in terms of a normal and position: [P:= (n,a) ] where (a ) is a point on the plane. The trick is then to find (a ) given just the normal and offset description. [n cdot x = w ] After looking at this for a minute, it appears that if I use the point: [a:= wn ] then [n cdot wn = w(n cdot n) = w ] so it is easy to find a point on the plane given the normal and offset.