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I'm interested in the proper way to give a mathematical definition of a certain geometric property exhibited by certain maps from points to sets.
Consider mappings from a n-dimensional space of real numbers P into subsets of an m-dimensional space S of real numbers.
For a practical application, you can think of a point in P as a vector of parameters that specifies the location, size, and orientation of a geometric figure. For example, it might be (x,y,theta,d) where (x,y) is the center of a square, theta is the orientation of the square relative to the horizontal axis and d is the length of a side of the square.
You can think of a mapping as a "rendering" program that takes a point in P as input and draws the geometric figure as a set of points in the space S. For example, S could be a 2-D screen where a square (including its interior) is drawn.
Assume we establish an n-dimensional grid in P. Let C be one cell of that grid. Imagine a rendering program that renders the figures corresponding to each point in C, including those on its boundary. Pretend that renders the figures so that, when it is finished, each point that was in at least one of the figures is colored blue.
There are examples we can establish the "outer" boundary of the blue area by rendering only the figures defined by points on the boundary of C. (Intuitively, the points on the boundary of C represent the most "extreme" positions of the figure defined by points in C, so this is not surprising.) I'm interested in formalizing this property or knowing it's name if it already has one.
Consider mappings from a n-dimensional space of real numbers P into subsets of an m-dimensional space S of real numbers.
For a practical application, you can think of a point in P as a vector of parameters that specifies the location, size, and orientation of a geometric figure. For example, it might be (x,y,theta,d) where (x,y) is the center of a square, theta is the orientation of the square relative to the horizontal axis and d is the length of a side of the square.
You can think of a mapping as a "rendering" program that takes a point in P as input and draws the geometric figure as a set of points in the space S. For example, S could be a 2-D screen where a square (including its interior) is drawn.
Assume we establish an n-dimensional grid in P. Let C be one cell of that grid. Imagine a rendering program that renders the figures corresponding to each point in C, including those on its boundary. Pretend that renders the figures so that, when it is finished, each point that was in at least one of the figures is colored blue.
There are examples we can establish the "outer" boundary of the blue area by rendering only the figures defined by points on the boundary of C. (Intuitively, the points on the boundary of C represent the most "extreme" positions of the figure defined by points in C, so this is not surprising.) I'm interested in formalizing this property or knowing it's name if it already has one.