I'm trying to calculate the values shown in the picture in red i.e. the interior angles.
I've got an array of the points where lines intersect and have tried using the dot-product but it only returns the smallest angles. I need the full range of internal angles (0-359) but can't seem to find much that meets this criteria.
Assuming your angles are in standard counterclockwise format, the following should work:
void angles(double points[][2], double angles[], int npoints){
for(int i = 0; i < npoints; i++){
int last = (i - 1 + npoints) % npoints;
int next = (i + 1) % npoints;
double x1 = points[i][0] - points[last][0];
double y1 = points[i][1] - points[last][1];
double x2 = points[next][0] - points[i][0];
double y2 = points[next][1] - points[i][1];
double theta1 = atan2(y1, x1)*180/3.1415926358979323;
double theta2 = atan2(y2, x2)*180/3.1415926358979323;
angles[i] = (180 + theta1 - theta2 + 360);
while(angles[i]>360)angles[i]-=360;
}
}
Obviously, if you are using some sort of data structure for your points, you will want to replace double points[][2] and references to it with references to your data structure.
You can obtain full angle range (-Pi..Pi) with atan2 function:
atan2(crossproduct, dotproduct)
Related
How can I draw a line starting from one point in the direction of the angle between the points?
This is what I have for calculating the angle
double angle1 = atan2(point_1_y - point_2_y, point_3_x - point_2_x);
double angle2 = atan2(point_1_y - point_3_y, point_3_x - point_2_x);
double result = angle1 - angle2;
First write a function to normalise a vector:
double mag = sqrt(x*x + y*y);
x = x/mag;
y = y/mag;
(I don't know what language you use, so I don't know the syntax of a function.)
Then apply it to the two vectors:
double x1 = point_C_x - point_A_x;
double y1 = point_C_y - point_A_y;
normalise(x1, y1);
double x2 = point_B_x - point_A_x;
double y2 = point_B_y - point_A_y;
normalise(x2, y2);
Then add them to get a green vector, then use atan2:
double x_green = x1+x2;
double y_green = y1+y2;
double d = atan2(y_green, x_green);
I'm writing a program that can draw a line between two points with filled circles. The circles:
- shouldn't overlap each other
- be as close together as possible
- and the centre of each circle should be on the line.
I've written a function to produce the circles, however I'm having trouble calculating position of each circle so that they are correctly lined up
void addCircles(scrPt endPt1, scrPt endPt2)
{
float xLength, yLength, length, cSquare, slope;
int numberOfCircles;
// Get the x distance between the two points
xLength = abs(endPt1.x - endPt2.x);
// Get the y distance between the two points
yLength = abs(endPt1.y - endPt2.y);
// Get the length between the points
cSquare = pow(xLength, 2) + pow(yLength, 2);
length = sqrt(cSquare);
// calculate the slope
slope = (endPt2.y - endPt1.y) / (endPt2.x - endPt1.x);
// Find how many circles fit inside the length
numberOfCircles = round(length / (radius * 2) - 1);
// set the position of each circle
for (int i = 0; i < numberOfCircles; i++)
{
scrPt circPt;
circPt.x = endPt1.x + ((radius * 2) * i);
circPt.y = endPt1.y + (((radius * 2) * i) * slope);
changeColor();
drawCircle (circPt.x, circPt.y);
}
This is what the above code produces:
I'm quite certain that the issue lies with this line, which sets the y value of the circle:
circPt.y = endPt1.y + (((radius * 2) * i) * slope);
Any help would be greatly appreciated
I recommend to calculate the direction of the line as a unit vector:
float xDist = endPt2.x - endPt1.x;
float yDist = endPt2.y - endPt1.y;
float length = sqrt(xDist*xDist + yDist *yDist);
float xDir = xDist / length;
float yDir = yDist / length;
Calculate the distance from one center point to the next one, numberOfSegments is the number of sections and not the number of circles:
int numberOfSegments = (int)trunc( length / (radius * 2) );
float distCpt = numberOfSegments == 0 ? 0.0f : length / (float)numberOfSegments;
A center point of a circle is calculated by the adding a vector the the start point of the line. The vector pints in the direction of the line and its length is given, by the distance between 2 circles multiplied by the "index" of the circle:
for (int i = 0; i <= numberOfSegments; i++)
{
float cpt_x = endPt1.x + xDir * distCpt * (float)i;
float cpt_y = endPt1.y + yDir * distCpt * (float)i;
changeColor();
drawCircle(cpt_x , cpt_y);
}
Note, the last circle on a line may be redrawn, by the first circle of the next line. You can change this by changing the iteration expression of the for loop - change <= to <:
for (int i = 0; i < numberOfSegments; i++)
In this case at the end of the line won't be drawn any circle at all.
I am attempting to implement Perlin Noise in c++.
Firstly, the problem (I think) is that the output is not what I expect. Currently I simply use the generated Perlin Noise values in a greyscaled image, and this is the results I get:
However, from my understanding, it's supposed to look more along the lines of:
That is, the noise I am producing currently seems to be more along the lines of "standard" irregular noise.
This is the Perlin Noise Algorithm I have implemented so far:
float perlinNoise2D(float x, float y)
{
// Find grid cell coordinates
int x0 = (x > 0.0f ? static_cast<int>(x) : (static_cast<int>(x) - 1));
int x1 = x0 + 1;
int y0 = (y > 0.0f ? static_cast<int>(y) : (static_cast<int>(y) - 1));
int y1 = y0 + 1;
float s = calculateInfluence(x0, y0, x, y);
float t = calculateInfluence(x1, y0, x, y);
float u = calculateInfluence(x0, y1, x, y);
float v = calculateInfluence(x1, y1, x, y);
// Local position in the grid cell
float localPosX = 3 * ((x - (float)x0) * (x - (float)x0)) - 2 * ((x - (float)x0) * (x - (float)x0) * (x - (float)x0));
float localPosY = 3 * ((y - (float)y0) * (y - (float)y0)) - 2 * ((y - (float)y0) * (y - (float)y0) * (y - (float)y0));
float a = s + localPosX * (t - s);
float b = u + localPosX * (v - u);
return lerp(a, b, localPosY);
}
The function calculateInfluence has the job of generating the random gradient vector and distance vector for one of the corner points of the current grid cell and returning the dot product of these. It is implemented as:
float calculateInfluence(int xGrid, int yGrid, float x, float y)
{
// Calculate gradient vector
float gradientXComponent = dist(rdEngine);
float gradientYComponent = dist(rdEngine);
// Normalize gradient vector
float magnitude = sqrt( pow(gradientXComponent, 2) + pow(gradientYComponent, 2) );
gradientXComponent = gradientXComponent / magnitude;
gradientYComponent = gradientYComponent / magnitude;
magnitude = sqrt(pow(gradientXComponent, 2) + pow(gradientYComponent, 2));
// Calculate distance vectors
float dx = x - (float)xGrid;
float dy = y - (float)yGrid;
// Compute dot product
return (dx * gradientXComponent + dy * gradientYComponent);
}
Here, dist is a random number generator from C++11:
std::mt19937 rdEngine(1);
std::normal_distribution<float> dist(0.0f, 1.0f);
And lerp is simply implemented as:
float lerp(float v0, float v1, float t)
{
return ( 1.0f - t ) * v0 + t * v1;
}
To implement the algorithm, I primarily made use of the following two resources:
Perlin Noise FAQ
Perlin Noise Pseudo Code
It's difficult for me to pinpoint exactly where I seem to be messing up. It could be that I am generating the gradient vectors incorrectly, as I'm not quite sure what type of distribution they should have. I have tried with a uniform distribution, however this seemed to generate repeating patterns in the texture!
Likewise, it could be that I am averaging the influence values incorrectly. It has been a bit difficult to discern exactly how it should be done from from the Perlin Noise FAQ article.
Does anyone have any hints as to what might be wrong with the code? :)
It seems like you are only generating a single octave of Perlin Noise. To get a result like the one shown, you need to generate multiple octaves and add them together. In a series of octaves, each octave should have a grid cell size double that of the last.
To generate multi-octave noise, use something similar to this:
float multiOctavePerlinNoise2D(float x, float y, int octaves)
{
float v = 0.0f;
float scale = 1.0f;
float weight = 1.0f;
float weightTotal = 0.0f;
for(int i = 0; i < octaves; i++)
{
v += perlinNoise2D(x * scale, y * scale) * weight;
weightTotal += weight;
// "ever-increasing frequencies and ever-decreasing amplitudes"
// (or conversely decreasing freqs and increasing amplitudes)
scale *= 0.5f;
weight *= 2.0f;
}
return v / weightTotal;
}
For extra randomness you could use a differently seeded random generator for each octave. Also, the weights given to each octave can be varied to adjust the aesthetic quality of the noise. If the weight variable is not adjusted each iteration, then the example above is "pink noise" (each doubling of frequency carries the same weight).
Also, you need to use a random number generator that returns the same value each time for a given xGrid, yGrid pair.
I'm trying to apply the Gauss-Bonnet theorem to my C++ OpenGL application and compute the value of the interior angle at vertex Vi in the neighboring triangle Fi in my mesh.
I did some searching before making this post, and I know that to do this for a 2D model, one could use the function below to get the angles:
void angles(double points[][2], double angles[], int npoints){
for(int i = 0; i < npoints; i++){
int last = (i - 1 + npoints) % npoints;
int next = (i + 1) % npoints;
double x1 = points[i][0] - points[last][0];
double y1 = points[i][1] - points[last][1];
double x2 = points[next][0] - points[i][0];
double y2 = points[next][1] - points[i][1];
double theta1 = atan2(y1, x1)*180/3.1415926358979323;
double theta2 = atan2(y2, x2)*180/3.1415926358979323;
angles[i] = (180 + theta1 - theta2 + 360);
while(angles[i]>360)angles[i]-=360;
} }
But how can I find the angles with a 3D mesh (x, y, and z) vertices?
The analogous concept in 3D is called Gaussian curvature. The situation is much more complicated than 2D, and there is no single good way of calculating or estimating the Gaussian curvature for a mesh. There's a survey paper that may give you some ideas.
I have been testing collision between two circles using the method:
Circle A = (x1,y1) Circle b = (x2,y2)
Radius A Radius b
x1 - x2 = x' * x'
y1 - y2 = y' * y'
x' + y' = distance
square root of distance - Radius A + Radius B
and if the resulting answer is a negative number it is intersecting.
I have used this method in a test but it doesn't seem to be very accurate at all.
bool circle::intersects(circle & test)
{
Vector temp;
temp.setX(centre.getX() - test.centre.getX());
temp.setY(centre.getY() - test.centre.getY());
float distance;
float temp2;
float xt;
xt = temp.getX();
temp2 = xt * xt;
temp.setX(temp2);
xt = temp.getY();
temp2 = xt * xt;
temp.setY(temp2);
xt = temp.getX() + temp.getY();
distance = sqrt(xt);
xt = radius + test.radius;
if( distance - xt < test.radius)
{
return true;
}
else return false;
}
This is the function using this method maybe I'm wrong here. I just wondered what other methods I could use. I know separating axis theorem is better , but I wouldn't know where to start.
if( distance - xt < test.radius)
{
return true;
}
distance - xt will evaluate to the blue line, the distance between the two disks. It also meets the condition of being less than the test radius, but there is no collision going on.
The solution:
if(distance <= (radius + test.radius) )
return true;
Where distance is the distance from the centres.
Given: xt = radius + test.radius;
The correct test is: if( distance < xt)
Here is an attempt to re-write the body for you: (no compiler, so may be errors)
bool circle::intersects(circle & test)
{
float x = this->centre.getX() - test.centre.getX()
float y = this->centre.getY() - test.centre.getY()
float distance = sqrt(x*x+y*y);
return distance < (this->radius + test.radius);
}
Based on Richard solution but comparing the squared distance. This reduce the computation errors and the computation time.
bool circle::intersects(circle & test)
{
float x = this->centre.getX() - test.centre.getX()
float y = this->centre.getY() - test.centre.getY()
float distance2 = x * x + y * y;
float intersect_distance2 = (this->radius + test.radius) * (this->radius + test.radius);
return distance <= intersect_distance2;
}
Use Pythagoras theorem to compute the distance between the centres
That is a straight line
If they have collided then that distance is shorter that the sum of the two radiuses