What is the optimized (avoiding square roots, trigonometry etc.) formula for hit testing given x,y point and ellipse with cx,cy center and width,height?
(Ellipse can rotate, but that's not the case.)
Related
My app captures the shape of a room by having the user point a camera at floor corners, and then doing a bunch of math, eventually ending up with a polygon.
The assumption is that the walls are straight (not curved). The majority of the corners are formed by walls at right angles to each other, but in some cases might not be.
Depending on how accurately the user points the camera, the (x,y) coordinates I derive for the corner might be beyond the actual corner, or in front of the actual camera, or, less likely, to the left or right. Obviously, in this case, when I connect the dots, I get weird parallelogram or rhomboid shapes. See example.
I am looking for a program or algorithm to normalize or regularize these shapes, provided we know which corners are supposed to be right angles.
My initial attempt involved finding segments which had angles which were "close" to each other, adjust them all to the same angle, and then recalculate the vertices. However, this algorithm proved to be unstable.
My current thinking is to find angles which are most obtuse (as would be caused by a point mistakenly placed beyond the actual corner), or most acute (as would be caused by a point mistakenly placed in front of the actual corner), and find the corner point which would make it a right angle. The problem, however, is that such as adjustment could have side-effects on other corners, such as making them even further away from right angles. I sense I need some kind of algorithm which takes all the information and optimizes/solves it at once--is this a kind of linear programming problem?--but I am stuck.
There is not a unique solution.
For example, take the perpendicular from the middle point of an edge to the two neighboring edges. This will give you two new corners.
Or take the perpendicular from the end point of an edge to other edges.
Or compute the average of angles in the end points of an edge. Use this average and the middle point of the edge to compute new corners.
Or...
To get the most faithful compliance, capture (or calculate) distances from each corner to the other three. Build triangles with those distances. Then use the average of the coordinates you compute for a corner from 2 or 3 triangles.
Resulting angles will not be exactly 90 degrees, but the polygon will represent the room fairly.
I have a straight line which intersects a convex polygon in 2D plane. There exists a circle with constant radius. The center of circle is moving on this line. So at first the polygon and circle don't intersect with each other, as the circle gets closer to the polygon the intersection increases and then decreases as they go further from each other. I want to prove the area of the intersection of the convex polygon and circle doesn't have local minima(as the circle moves on the line).
Interesting problem. Please post solution once you find it. My approach would be to take a similar route to Fortunes algorithm to build a Voronoi graph - meaning I would consider "events" that are happening when the circle traverses a convex polygon.
Basically to better understand the problem, consider the restriction that the circle is traveling on straight line - why is that important - look at counter examples. Then look when will this fail if poly is not convex?
The events that I would consider would be an entry/exit of a poly vertex into circle, and entry exit of an poly edge from/into the circle. Then keep track of area increasing or decreasing through each event, and show that it is necessarily monotonic.
I came across this link http://www.mathopenref.com/coordpolygonarea2.html
It explains how to calculate the area of a polygon and helps to identify whether the polygon vertices we entered is clockwise or counter clockwise.
If area value is +ve, it is clockwise, if it is -nv then it is in counterclockwise.
My requirement is to identify only whether it is clockwise or counterclockwise. Is this rule will work correctly (though there are limitations as mentioned in the link). I have only regular polygons (not complicated, no self intersections) but the vertices are more.
I am not interested in the area value accuracy, just to know the ring rotation.
Any other thought on this.
For convex polygons:
Select two edges with a common vertex.
Lets say, edge1 is between vertex A and B. Edge2 is between vertex B and C.
Define to vectors: vect1: A----->B
vect2: B----->C
Cross product vect1 and vect2.
If the result is positive, the sequence A-->B-->C is Counter-clockwise.
If the result is negative, the sequence A-->B-->C is clockwise.
If you have only convex polygons (and all regular polygons are convex), and if your points are all organized consistently--either all counterclockwise or all clockwise--then you can determine which by just computing the (signed) area of one triangle determined by any three consecutive points. This is essentially computing the cross product of the two vectors along the two edges.
Suppose there's a set of 2D points to represent an initial simple polygon. Now I want to optimize the positions of each point according to some cost function. But this could make the polygon complex, i.e. the polygon intersects with itself. How can I avoid this? Thanks!
If the polygon could be presumed to be convex, then it is simple. Simply compute the angles between each side and the next side. Each angle must be between 0 and 180 degrees for a convex polygon. The sum of those angles is well known for a closed polygon with N sides. This will result in a simple constrained optimization. (Actually, you can write those constraints in a "simpler" form than computing the angles with a trigonometric function. Cross products will suffice.)
If the polygon need not be convex, then you need to worry about edges crossing, or other degeneracies.
How can I calculate the size of a circle from a set of arcs?
Specifically, I have this SVG path definition which draws a circle, I'm looking to work out its size.
<path clip-path="url(#SVGID_2_)" fill="#99C44C" d="M334.293,56.846c0-4.782,3.88-8.659,8.665-8.659c4.78,0,8.66,3.877,8.66,8.659
c0,4.783-3.88,8.661-8.66,8.661C338.173,65.507,334.293,61.629,334.293,56.846"/>
For your information, the circle is drawn in the 'd' attribute. M334.293,56.846 moves to this x,y position, then the c commands are curves.
Curves:
Draws a cubic Bézier curve from the current point to (x,y) using (x1,y1) as the control point at the beginning of the curve and (x2,y2) as the control point at the end of the curve. C (uppercase) indicates that absolute coordinates will follow; c (lowercase) indicates that relative coordinates will follow.
relative curves
c0-4.782,3.88-8.659,8.665-8.659
c4.78,0,8.66,3.877,8.66,8.659
c0,4.783-3.88,8.661-8.66,8.661
absolute curve
C338.173,65.507,334.293,61.629,334.293,56.846
At this juncture you have two possibilites:
You can treat the bezier curve as a circle (which is, as commented, wrong; it's just really circle-looking). To calculate the area, determine the radius and use π * r^2 as usual.
If you want to calculate generally the area enclosed by a path element; that requires some moderate calculus, and is not for the faint of heart.