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approximation.md

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+# Approximation
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+Approximating means calculating something with lesser than best possible precision -- estimating -- purposefully allowing some margin of error in results and using simpler mathematical models than the most accurate ones: this is typically done in order to save resources (CPU cycles, memory etc.) and reduce [complexity](complexity.md) so that our projects stay manageable. Simulating real world on a computer is always an approximation as we cannot capture the infinitely complex and fine nature of the real world with a machine of limited resources, but even withing this we need to consider how much, in what ways and where to simplify.
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+Using approximations however doesn't have to imply decrease in precision of the final result -- approximations very well serve [optimization](optimization.md). E.g. approximate metrics help in [heuristic](heuristic.md) algorithms such as [A*](a_star.md). Another use of approximations in optimization is as a quick preliminary check for the expensive precise algorithms: e.g. using bounding spheres helps speed up collision detection (if bounding spheres of two objects don't collide, we know they can't possibly collide and don't have to expensively check this).
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+Example of approximations:
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+- **Distances**: instead of expensive **Euclidean** distance (`sqrt(dx^2 + dy^2)`) we may use **Chebyshev** distance (`dx + dy`) or **Taxicab** distance (`max(dx,dy)`).
+- **Engineering approximations**: e.g. **sin(x) = x** for "small" values of *x* or **pi = 3** (integer instead of float).
+- **Physics engines**: complex triangle meshes are approximated with simple analytical shapes such as **spheres**, **cuboids** and **capsules** or at least **convex hulls** which are much easier and faster to deal with. They also approximate **relativistic** physics with **Newtonian**.
+- **Real time graphics engines**, on the other hand, normally approximate all shapes with triangle meshes.
+- **[Ray tracing](ray_tracing.md)** neglects indirect lighting. Computer graphics in general is about approximating the solution of the rendering equation.
+- **Real numbers** are practically always approximated with [floating point](floating_point.md) or [fixed point](fixed_point.md) (rational numbers).
+- **[Numerical methods](numerical.md)** offer generality and typically yield approximate solutions while their precision vs speed can be adjusted via parameters such as number of iterations.
+- **[Taylor series](taylor_series.md)** approximates given mathematical function and can be used to e.g. estimate solutions of [differential equations](differential_equation.md).