I have this seemingly simple but very confusing problem.
Given I have a set of vertices (x1,y1), (x2,y2), (x3,y3)...... representing an arc. The points can either be clockwise or counter clockwise, but are all similarly ordered.
And I know the center of the arc (xc,yc).
How can I tell if the arc subtends an acute/obtuse or reflex angle?
One obvious solution is to take the difference of atan2((last_pt)-(center)) and atan2((first_pt)-(center))). But if the arc goes through the point where PI become -PI, this method breaks down.
Also, since the arc points are derived from a rather noisy pixelated picture, the vertices are not exactly smooth.
Picture of a acute and reflex arc
I cant wrap my brain around solving this problem.
Thanks for your help!
Working with 2D angles is a pain for the reason you described, so it's better to work with vector math instead, which is rotationally invariant.
Define the 2D cross-product, A ^ B = Ax * By - Ay * Bx. This is positive if A is clockwise rotated relative to B, and vice versa.
The logic:
Compute C = (last_pt - center) ^ (first_pt - center)
If C = 0, the arc is either closed or 180-degree (forgot the name for this)
If C > 0, the arc must either be (i) clockwise and acute/obtuse or (ii) anti-clockwise and reflex
If C < 0, the opposite applies
Pseudocode:
int arc_type(Point first, Point last, Point center, bool clockwise)
{
// cross-product
float C = (last.x - center.x) * (first.y - center.y)
- (last.y - center.y) * (first.x - center.x);
if (Math.abs(C) < /* small epsilon */)
return 0; // 180-degree
return ((C > 0) ^ clockwise) ? 1 // reflex
: -1; // acute / obtuse
}
Note that if you don't have prior knowledge of whether the arc is clockwise or anti-, you can use the same cross-product method on adjacent points. You need to ensure that the order of the points is consistent - if not you can, again using the cross-product, sort them by (relative) angle.
Related
Question:
I need to calculate intersection shape (purple) of plane defined by Ax + By + Cz + D = 0 and frustum defined by 4 rays emitting from corners of rectangle (red arrows). The result shoud be quadrilateral (4 points) and important requirement is that result shape must be in plane's local space. Plane is created with transformation matrix T (planes' normal is vec3(0, 0, 1) in T's space).
Explanation:
This is perspective form of my rectangle projection to another space (transformation / matrix / node). I am able to calculate intersection shape of any rectangle without perspective rays (all rays are parallel) by plane-line intersection algorithm (pseudocode):
Definitions:
// Plane defined by normal (A, B, C) and D
struct Plane { vec3 n; float d; };
// Line defined by 2 points
struct Line { vec3 a, b; };
Intersection:
vec3 PlaneLineIntersection(Plane plane, Line line) {
vec3 ba = normalize(line.b, line.a);
float dotA = dot(plane.n, l.a);
float dotBA = dot(plane.n, ba);
float t = (plane.d - dotA) / dotBA;
return line.a + ba * t;
}
Perspective form comes with some problems, because some of rays could be parallel with plane (intersection point is in infinite) or final shape is self-intersecting. Its works in some cases, but it's not enough for arbitary transformation. How to get correct intersection part of plane wtih perspective?
Simply, I need to get visible part of arbitary plane by arbitary perspective "camera".
Thank you for suggestions.
Intersection between a plane (one Ax+By+Cx+D equation) and a line (two planes equations) is a matter of solving the 3x3 matrix for x,y,z.
Doing all calculations on T-space (origin is at the top of the pyramid) is easier as some A,B,C are 0.
What I don't know if you are aware of is that perspective is a kind of projection that distorts the z ("depth", far from the origin). So if the plane that contains the rectangle is not perpendicular to the axis of the fustrum (z-axis) then it's not a rectangle when projected into the plane, but a trapezoid.
Anyhow, using the projection perspective matrix you can get projected coordinates for the four rectangle corners.
To tell if a point is in one side of a plane or in the other just put the point coordinates in the plane equation and get the sign, as shown here
Your question seems inherently mathematic so excuse my mathematical solution on StackOverflow. If your four arrows emit from a single point and the formed side planes share a common angle, then you are looking for a solution to the frustum projection problem. Your requirements simplify the problem quite a bit because you define the plane with a normal, not two bounded vectors, thus if you agree to the definitions...
then I can provide you with the mathematical solution here (Internet Explorer .mht file, possibly requiring modern Windows OS). If you are thinking about an actual implementation then I can only direct you to a very similar frustum projection implementation that I have implemented/uploaded here (Lua): https://github.com/quiret/mta_lua_3d_math
The roadmap for the implementation could be as follows: creation of condition container classes for all sub-problems (0 < k1*a1 + k2, etc) plus the and/or chains, writing algorithms for the comparisions across and-chains as well as normal-form creation, optimization of object construction/memory allocation. Since each check for frustum intersection requires just a fixed amount of algebraic objects you can implement an efficient cache.
I know the formula to know if a point is inside, outside and on a circle : https://math.stackexchange.com/q/198769 This quote explains that we must compare d to r (please read the quote, it's only 5 lines).
But I just want to know if a point is ON a circle. Moreover, and that's the real problem : if a point is a bit inside/outside a circle, I want to consider it as ON the circle.
How could I do that ? I tried to delimit d-r (ie. : the comparison) in a range. Example :
if(d-r > -100 && d-r < 100) { point is on the circle }
It works, with -100 and 100, for circles with a little radius (ie. : ALL the points that are a bit outside/inside the circle are considered as being on the circle).
But for circles for a big radius, only SOME points are considered as being on the circle (ie. : only some of the points that are a bit outside/inside the circle are considered as being on the circle)...
So I would want that ALL the points that are a bit outside/inside the circle are considered as being on the circle, independently of the circle's radius. How ?
Your comparison for absolute difference might be written shorter as
if Abs(d - r) < delta (i.e. 100) ...
But seems you need relative difference depending on circle radius like this:
if Abs(d - r) / r < reldelta (i.e. 0.001) ...
From a probabilistic perspective, you could define a sort of distance map (as proposed by #Mbo) adopting the relative distance and use it to build a probability distribution on each point. The probability would represent a sort of likelihood of the point to belong to the circle. Intuitively, the closer the point, the more likely it is to be part of the circle. For example:
rel_d = (d-r)/r;
// P(x on the circle) = 1 - rel_d
if(rel_d < 1){
P_on_circle = 1 - rel_d;
}else{
P_on_circle = 0;
}
I've been looking at and trying to understand the following bit of code
float sdBox( vec3 p, vec3 b )
{
vec3 d = abs(p) - b;
return min(max(d.x,max(d.y,d.z)),0.0) +
length(max(d,0.0));
}
I understand that length(d) handles the SDF case where the point is off to the 'corner' (ie. all components of d are positive) and that max(d.x, d.y, d.z) gives us the proper distance in all other cases. What I don't understand is how these two are combined here without the use of an if statement to check the signs of d's components.
When all of the d components are positive, the return expression can be reduced to length(d) because of the way min/max will evaluate - and when all of the d components are negative, we get max(d.x, d.y, d.z). But how am I supposed to understand the in-between cases? The ones where the components of d have mixed signs?
I've been trying to graph it out to no avail. I would really appreciate it if someone could explain this to me in geometrical/mathematical terms. Thanks.
If you like to know how It works It's better do the following steps:
1.first of all you should know definitions of shapes
2.It's always better to consider 2D shape of them, because three dimensions may be complex for you.
so let me to explain some shapes:
Circle
A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the center.
You can use distance(), length() or sqrt() to calculate the distance to the center of the billboard.
The book of shaders - Chapter 7
Square
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles).
I describe 2D shapes In before section now let me to describe 3D definition.
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.
Like a circle, which geometrically is an object in two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in three-dimensional space.
Refrence - Wikipedia
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
Refrence : Wikipedia
Modeling with distance functions
now It's time to understanding modeling with distance functions
Sphere
As mentioned In last sections.In below code length() used to calculate the distance to the center of the billboard , and you can scale this shape by s parameter.
//Sphere - signed - exact
/// <param name="p">Position.</param>
/// <param name="s">Scale.</param>
float sdSphere( vec3 p, float s )
{
return length(p)-s;
}
Box
// Box - unsigned - exact
/// <param name="p">Position.</param>
/// <param name="b">Bound(Scale).</param>
float udBox( vec3 p, vec3 b )
{
return length(max(abs(p)-b,0.0));
}
length() used like previous example.
next we have max(x,0) It called Positive and negative parts
this is mean below code is equivalent:
float udBox( vec3 p, vec3 b )
{
vec3 value = abs(p)-b;
if(value.x<0.){
value.x = 0.;
}
if(value.y<0.){
value.y = 0.;
}
if(value.z<0.){
value.z = 0.;
}
return length(value);
}
step 1
if(value.x<0.){
value.x = 0.;
}
step 2
if(value.y<0.){
value.y = 0.;
}
step 3
if(value.z<0.){
value.z = 0.;
}
step 4
next we have absolution function.It used to remove additional parts.
Absolution Steps
Absolution step 1
if(value.x < -1.){
value.x = 1.;
}
Absolution step 2
if(value.y < -1.){
value.y = 1.;
}
Absolution step 3
if(value.z < -1.){
value.z = 1.;
}
Also you can make any shape by using Constructive solid geometry.
CSG is built on 3 primitive operations: intersection ( ∩ ), union ( ∪ ), and difference ( - ).
It turns out these operations are all concisely expressible when combining two surfaces expressed as SDFs.
float intersectSDF(float distA, float distB) {
return max(distA, distB);
}
float unionSDF(float distA, float distB) {
return min(distA, distB);
}
float differenceSDF(float distA, float distB) {
return max(distA, -distB);
}
I figured it out a while ago and wrote about this extensively in a blog post here: http://fabricecastel.github.io/blog/2016-02-11/main.html
Here's an excerpt (see the full post for a full explanation):
Consider the four points, A, B, C and D. Let's crudely reduce the distance function to try and get rid of the min/max functions in order to understand their effect (since that's what's puzzling about this function). The notation below is a little sloppy, I'm using square brackets to denote 2D vectors.
// 2D version of the function
d(p) = min(max(p.x, p.y), 0)
+ length(max(p, 0))
---
d(A) = min(max(-1, -1), 0)
+ length(max([-1, -1], 0))
d(A) = -1 + length[0, 0]
---
d(B) = min(max(1, 1), 0)
+ length(max([1, 1], 0))
d(B) = 0 + length[1, 1]
Ok, so far nothing special. When A is inside the square, we essentially get our first distance function based on planes/lines and when B is in the area where our first distance function is inaccurate, it gets zeroed out and we get the second distance function (the length). The trick lies in the other two cases C and D. Let's work them out.
d(C) = min(max(-1, 1), 0)
+ length(max([-1, 1], 0))
d(C) = 0 + length[0, 1]
---
d(D) = min(max(1, -1), 0)
+ length(max([-1, 1], 0))
d(D) = 0 + length[1, 0]
If you look back to the graph above, you'll note C' and D'. Those points have coordinates [0,1] and [1,0], respectively. This method uses the fact that both distance fields intersect on the axes - that D and D' lie at the same distance from the square.
If we zero out all negative component of a vector and take its length we will get the proper distance between the point and the square (for points outside of the square only). This is what max(d,0.0) does; a component-wise max operation. So long as the vector has at least one positive component, min(max(d.x,d.y),0.0) will resolve to 0 leaving us with only the second part of the equation. In the event that the point is inside the square, we want to return the first part of the equation (since it represents our first distance function). If all components of the vector are negative it's easy to see our condition will be met.
This understanding should tranlsate back into 3D seamlessly once you wrap your head around it. You may or may not have to draw a few graphs by hand to really "get" it - I know I did and would encourage you to do so if you're dissatisfied with my explanation.
Working this implementation into our own code, we get this:
float distanceToNearestSurface(vec3 p){
float s = 1.0;
vec3 d = abs(p) - vec3(s);
return min(max(d.x, max(d.y,d.z)), 0.0)
+ length(max(d,0.0));
}
And there you have it.
I'm trying to find the best way to get the most distant point of a circle from a specified point in 2D space. What I have found so far, is how to get the distance between the point and the circle position, but I'm not entirely sure how to expand this to find the most distant point of the circle.
The known variables are:
Point a
Point b (circle position)
Radius r (circle radius)
To find the distance between the point and the circle position, I have found this:
xd = x2 - x1
yd = y2 - y1
Distance = SquareRoot(xd * xd + yd * yd)
It seems to me, this is part of the solution. How would this be expanded to get the position of Point x in the below image?
As an additional but optional part of the question: I have read in some places that it would be possible to get the distance portion without using the Square Root, which is very performance intensive and should be avoided if fast code is necessary. In my case, I would be doing this calculation quite often; Any comments on this within the context of the main question would be welcome too.
What about this?
Calculate A-B.
We now have a vector pointing from the center of the circle towards A (if B is the origin, skip this and just consider point A a vector).
Normalize.
Now we have a well defined length (the length is 1)
If the circle is not of unit radius, multiply by radius. If it is unit radius, skip this.
Now we have the correct length.
Invert sign (can be done in one step with 3., just multiply with the negative radius)
Now our vector points in the correct direction.
Add B (if B is the origin, skip this).
Now our vector is offset correctly so its endpoint is the point we want.
(Alternatively, you could calculate B-A to save the negation, but then you have to do one more operation to offset the origin correctly.)
By the way, it works the same in 3D, except the circle would be a sphere, and the vectors would have 3 components (or 4, if you use homogenous coords, in this case remember -- for correctness -- setting w to 0 when "turning points into vectors" and to 1 at the end when making a point from the vector).
EDIT:
(in reply of pseudocode)
Assuming you have a vec2 class which is a struct of two float numbers with operators for vector subtraction and scalar multiplicaion (pretty trivial, around a dozen lines of code) and a function normalize which needs to be no more than a shorthand for multiplying with inv_sqrt(x*x+y*y), the pseudocode (my pseudocode here is something like a C++/GLSL mix) could look something like this:
vec2 most_distant_on_circle(vec2 const& B, float r, vec2 const& A)
{
vec2 P(A - B);
normalize(P);
return -r * P + B;
}
Most math libraries that you'd use should have all of these functions and types built-in. HLSL and GLSL have them as first type primitives and intrinsic functions. Some GPUs even have a dedicated normalize instruction.
OK, I know this sounds like it should be asked on math.stackoverflow.com, but this is embarrassingly simple maths that I've forgotten from high-school, rather than advanced post-graduate stuff!
I'm doing some graphics programming, and I have a triangle. Incidentally, two of this triangle's sides are equal, but I'm not sure if that's relevant. I have the coordinates of two of the corners (vertices), but not the third (these coordinates are pixels on the screen, in case that's relevant). I know the lengths of all three sides.
How do I determine the coordinates of the unknown vertex?
for oblique triangles: c^2 = a^2 + b^2 - 2ab * cos(C)
where a, b, c are the lengths of the sides (regardless of length)
and A, B, C are the angles opposite the side with the same letter.
Use the above to figure out the angle from one of the endpoints you know, then use the angle, the position of the vertex, and the angle between the adjacent sides to determine where the unknown vertex is.
And the complexity of the problem doesn't determine which site it should go on, only the subject matter. So you should move this to math.
EDIT: I had a serious brainfart previously, but this should work.
Use the law of cosines
/* use the law of cosines to get the angle of CAB */
c² = a² + b² - 2ab cos(Cangle)
cos(Cangle) = (a²+b²-c²) / 2ab
Cangle = acos((a²+b²-c²) / 2ab)
AB = B.xy - A.xy;
normalize(AB);
len = length(AC)
C.x = len*AB.x* cos(Cangle) * len*AB.y*sin(Cangle);
C.y = len*AB.x*-sin(Cangle) * len*AB.y*cos(Cangle);