Finally, we relate our discussion to the framework of path integral optimization by generalizing the original argument to non-trivial topologies. In particular, we show that the minimization procedure defining the complexity naturally leads to regular continued fractions, allowing an interesting geometric interpretation of the results in the Farey tesselation of the upper-half plane. The calculations present interesting number theoretic features related with continued fraction representations of rational numbers. We also use the braid word presentation of knots to discuss states on the four-punctured sphere Hilbert space associated with 2-bridge knots and links. The results are then generalized for a family of multi-component links that are obtained by "Hopf-linking" different torus knots. These can be constructed from the unknot state by using the Hilbert space representation of the $S$ and $T$ modular transformations of the torus as fundamental gates. Specifically, we deal with states in the torus Hilbert space of Chern-Simons that are the knot complements on the $3$-sphere of arbitrary torus knots. We compute the circuit complexity of quantum states in $3d$ Chern-Simons theory corresponding to certain classes of knots. ![]() ![]() In particular, we show that the minimization procedure defining the complexity naturally leads to regular continued fractions, allowing a geometric interpretation of the results in the Farey tesselation of the upper-half plane. Recognize the format of a polar rectangular regions double integral. We also use the braid word presentation of knots to discuss states on the punctured sphere Hilbert space associated with 2-bridge knots and links. notice that will sweep between the two intersection points of the two circles. The results are then generalized for a family of multi-component links that are obtained by “Hopf-linking” different torus knots. The upper bound is saturated in the semiclassical limit of Chern-Simons theory. These can be constructed from the unknot state by using the Hilbert space representation of the S and T modular transformations of the torus as fundamental gates. Specifically, we deal with states in the torus Hilbert space of Chern-Simons that are the knot complements on the 3-sphere of arbitrary torus knots. We compute an upper bound on the circuit complexity of quantum states in 3 d Chern-Simons theory corresponding to certain classes of knots. At the end of the paper, we note a connection between our work and the notion of extending geodesics. We are able to show that, with appropriate restrictions, the efficient geodesic algorithm can be used to build an algorithm that reduces i(v, w) while preserving d(v, w). So, combining the above two snippets yields you a 3-line function for circle-rectangle check:ĭeltaX = CircleX - Max(RectX, Min(CircleX, RectX RectWidth)) ĭeltaY = CircleY - Max(RectY, Min(CircleY, RectY RectHeight)) Īnd here's it in action, along with a bit of debug drawing:Īnd that is it.In this work, we study the cellular decomposition of S induced by a filling pair of curves v and w, \(Dec_(S)\) called spirals. NearestY = Max(RectY, Min(CircleY, RectY RectHeight)) What could be the maximum radius of two identical circles which can be completely fit inside a rectangle of dimension 3cm 4cm without overlappingFor more. NearestX = Max(RectX, Min(CircleX, RectX RectWidth)) Surprisingly or not, rectangle-circle collisions are not all too different - first you find the point of rectangle that is the closest to the circle' center, and check that point is in the circle.Īnd, if the rectangle is not rotated, finding a point closest to the circle' center is simply a matter of clamping the circle' center coordinates to rectangle coordinates: Return (DeltaX * DeltaX DeltaY * DeltaY ) < (CircleRadius * CircleRadius ) ![]() This is a blog post about handling circle-rectangle collisions.įor some reason, these seem to be generally regarded as something complicated, even though they aren't.įirst things first, you may already know how to check circle-point collision - it's simply checking that the distance between the circle' center and the point is smaller than the circle' radius:
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