Here is the code for an oval drawing method I am working on. I am applying the Bresenham method to plot its co-ordinates, and taking advantage of the ellipse's symmetrical properties to draw the same pixel in four different places.
void cRenderClass::plotEllipse(int xCentre, int yCentre, int width, int height, float angle, float xScale, float yScale)
{
if ((height == width) && (abs(xScale - yScale) < 0.005))
plotCircle(xCentre, yCentre, width, xScale);
std::vector<std::vector <float>> rotate;
if (angle > 360.0f)
{
angle -= 180.0f;
}
rotate = maths.rotateMatrix(angle, 'z');
//rotate[0][0] = cos(angle)
//rotate[0][1] = sin(angle)
float theta = atan2(-height*rotate[0][1], width*rotate[0][0]);
if (angle > 90.0f && angle < 180.0f)
{
theta += PI;
}
//add scalation in at a later date
float xShear = (width * (cos(theta) * rotate[0][0])) - (height * (sin(theta) * rotate[0][1]));
float yShear = (width * (cos(theta) * rotate[0][1])) + (height * (sin(theta) * rotate[0][0]));
float widthAxis = abs(sqrt(((rotate[0][0] * width) * (rotate[0][0] * width)) + ((rotate[0][1] * height) * (rotate[0][1] * height))));
float heightAxis = (width * height) / widthAxis;
int aSquared = widthAxis * widthAxis;
int fourASquared = 4*aSquared;
int bSquared = heightAxis * heightAxis;
int fourBSquared = 4*bSquared;
x0 = 0;
y0 = heightAxis;
int sigma = (bSquared * 2) + (aSquared * (1 - (2 * heightAxis)));
while ((bSquared * x0) <= (aSquared * y0))
{
drawPixel(xCentre + x0, yCentre + ((floor((x0 * yShear) / xShear)) + y0));
drawPixel(xCentre - x0, yCentre + ((floor((x0 * yShear) / xShear)) + y0));
drawPixel(xCentre + x0, yCentre + ((floor((x0 * yShear) / xShear)) - y0));
drawPixel(xCentre - x0, yCentre + ((floor((x0 * yShear) / xShear)) - y0));
if (sigma >= 0)
{
sigma += (fourASquared * (1 - y0));
y0--;
}
sigma += (bSquared * ((4 * x0) + 6));
x0++;
}
x0 = widthAxis;
y0 = 0;
sigma = (aSquared * 2) + (bSquared * (1 - (2 * widthAxis)));
while ((aSquared * y0) <= (bSquared * x0))
{
drawPixel(xCentre + x0, yCentre + ((floor((x0 * yShear) / xShear)) + y0));
drawPixel(xCentre - x0, yCentre + ((floor((x0 * yShear) / xShear)) + y0));
drawPixel(xCentre + x0, yCentre + ((floor((x0 * yShear) / xShear)) - y0));
drawPixel(xCentre - x0, yCentre + ((floor((x0 * yShear) / xShear)) - y0));
if (sigma >= 0)
{
sigma += (fourBSquared * (1 - x0));
x0--;
}
sigma += (aSquared * (4 * y0) + 6);
y0++;
}
//the above algorithm hasn't been quite completed
//there are still a few things I want to enquire Andy about
//before I move on
//this other algorithm definitely works
//however
//it is computationally expensive
//and the line drawing isn't as refined as the first one
//only use this as a last resort
/* std::vector<std::vector <float>> rotate;
rotate = maths.rotateMatrix(angle, 'z');
float s = rotate[0][1];
float c = rotate[0][0];
float ratio = (float)height / (float)width;
float px, py, xNew, yNew;
for (int theta = 0; theta <= 360; theta++)
{
px = (xCentre + (cos(maths.degToRad(theta)) * (width / 2))) - xCentre;
py = (yCentre - (ratio * (sin(maths.degToRad(theta)) * (width / 2)))) - yCentre;
x0 = (px * c) - (py * s);
y0 = (px * s) + (py * c);
drawPixel(x0 + xCentre, y0 + yCentre);
}*/
}
Here's the problem. When testing the rotation matrix on my oval drawing function, I expect it to draw an ellipse at a slant from its original horizontal position as signified by 'angle'. Instead, it makes a heart shape. This is sweet, but not the result I want.
I have managed to get the other algorithm (as seen in the bottom part of that code sample) working successfully, but it takes more time to compute, and doesn't draw lines quite as nicely. I only plan to use that if I can't get this Bresenham one working.
Can anyone help?
I have an ellipse, defined by Center Point, radiusX and radiusY, and I have a Point. I want to find the point on the ellipse that is closest to the given point. In the illustration below, that would be S1.
Now I already have code, but there is a logical error somewhere in it, and I seem to be unable to find it. I broke the problem down to the following code example:
#include <vector>
#include <opencv2/core/core.hpp>
#include <opencv2/highgui/highgui.hpp>
#include <math.h>
using namespace std;
void dostuff();
int main()
{
dostuff();
return 0;
}
typedef std::vector<cv::Point> vectorOfCvPoints;
void dostuff()
{
const double ellipseCenterX = 250;
const double ellipseCenterY = 250;
const double ellipseRadiusX = 150;
const double ellipseRadiusY = 100;
vectorOfCvPoints datapoints;
for (int i = 0; i < 360; i+=5)
{
double angle = i / 180.0 * CV_PI;
double x = ellipseRadiusX * cos(angle);
double y = ellipseRadiusY * sin(angle);
x *= 1.4;
y *= 1.4;
x += ellipseCenterX;
y += ellipseCenterY;
datapoints.push_back(cv::Point(x,y));
}
cv::Mat drawing = cv::Mat::zeros( 500, 500, CV_8UC1 );
for (int i = 0; i < datapoints.size(); i++)
{
const cv::Point & curPoint = datapoints[i];
const double curPointX = curPoint.x;
const double curPointY = curPoint.y * -1; //transform from image coordinates to geometric coordinates
double angleToEllipseCenter = atan2(curPointY - ellipseCenterY * -1, curPointX - ellipseCenterX); //ellipseCenterY * -1 for transformation to geometric coords (from image coords)
double nearestEllipseX = ellipseCenterX + ellipseRadiusX * cos(angleToEllipseCenter);
double nearestEllipseY = ellipseCenterY * -1 + ellipseRadiusY * sin(angleToEllipseCenter); //ellipseCenterY * -1 for transformation to geometric coords (from image coords)
cv::Point center(ellipseCenterX, ellipseCenterY);
cv::Size axes(ellipseRadiusX, ellipseRadiusY);
cv::ellipse(drawing, center, axes, 0, 0, 360, cv::Scalar(255));
cv::line(drawing, curPoint, cv::Point(nearestEllipseX,nearestEllipseY*-1), cv::Scalar(180));
}
cv::namedWindow( "ellipse", CV_WINDOW_AUTOSIZE );
cv::imshow( "ellipse", drawing );
cv::waitKey(0);
}
It produces the following image:
You can see that it actually finds "near" points on the ellipse, but it are not the "nearest" points. What I intentionally want is this: (excuse my poor drawing)
would you extent the lines in the last image, they would cross the center of the ellipse, but this is not the case for the lines in the previous image.
I hope you get the picture. Can anyone tell me what I am doing wrong?
Consider a bounding circle around the given point (c, d), which passes through the nearest point on the ellipse. From the diagram it is clear that the closest point is such that a line drawn from it to the given point must be perpendicular to the shared tangent of the ellipse and circle. Any other points would be outside the circle and so must be further away from the given point.
So the point you are looking for is not the intersection between the line and the ellipse, but the point (x, y) in the diagram.
Gradient of tangent:
Gradient of line:
Condition for perpedicular lines - product of gradients = -1:
When rearranged and substituted into the equation of your ellipse...
...this will give two nasty quartic (4th-degree polynomial) equations in terms of either x or y. AFAIK there are no general analytical (exact algebraic) methods to solve them. You could try an iterative method - look up the Newton-Raphson iterative root-finding algorithm.
Take a look at this very good paper on the subject:
http://www.spaceroots.org/documents/distance/distance-to-ellipse.pdf
Sorry for the incomplete answer - I totally blame the laws of mathematics and nature...
EDIT: oops, i seem to have a and b the wrong way round in the diagram xD
There is a relatively simple numerical method with better convergence than Newtons Method. I have a blog post about why it works http://wet-robots.ghost.io/simple-method-for-distance-to-ellipse/
This implementation works without any trig functions:
def solve(semi_major, semi_minor, p):
px = abs(p[0])
py = abs(p[1])
tx = 0.707
ty = 0.707
a = semi_major
b = semi_minor
for x in range(0, 3):
x = a * tx
y = b * ty
ex = (a*a - b*b) * tx**3 / a
ey = (b*b - a*a) * ty**3 / b
rx = x - ex
ry = y - ey
qx = px - ex
qy = py - ey
r = math.hypot(ry, rx)
q = math.hypot(qy, qx)
tx = min(1, max(0, (qx * r / q + ex) / a))
ty = min(1, max(0, (qy * r / q + ey) / b))
t = math.hypot(ty, tx)
tx /= t
ty /= t
return (math.copysign(a * tx, p[0]), math.copysign(b * ty, p[1]))
Credit to Adrian Stephens for the Trig-Free Optimization.
Here is the code translated to C# implemented from this paper to solve for the ellipse:
http://www.geometrictools.com/Documentation/DistancePointEllipseEllipsoid.pdf
Note that this code is untested - if you find any errors let me know.
//Pseudocode for robustly computing the closest ellipse point and distance to a query point. It
//is required that e0 >= e1 > 0, y0 >= 0, and y1 >= 0.
//e0,e1 = ellipse dimension 0 and 1, where 0 is greater and both are positive.
//y0,y1 = initial point on ellipse axis (center of ellipse is 0,0)
//x0,x1 = intersection point
double GetRoot ( double r0 , double z0 , double z1 , double g )
{
double n0 = r0*z0;
double s0 = z1 - 1;
double s1 = ( g < 0 ? 0 : Math.Sqrt(n0*n0+z1*z1) - 1 ) ;
double s = 0;
for ( int i = 0; i < maxIter; ++i ){
s = ( s0 + s1 ) / 2 ;
if ( s == s0 || s == s1 ) {break; }
double ratio0 = n0 /( s + r0 );
double ratio1 = z1 /( s + 1 );
g = ratio0*ratio0 + ratio1*ratio1 - 1 ;
if (g > 0) {s0 = s;} else if (g < 0) {s1 = s ;} else {break ;}
}
return s;
}
double DistancePointEllipse( double e0 , double e1 , double y0 , double y1 , out double x0 , out double x1)
{
double distance;
if ( y1 > 0){
if ( y0 > 0){
double z0 = y0 / e0;
double z1 = y1 / e1;
double g = z0*z0+z1*z1 - 1;
if ( g != 0){
double r0 = (e0/e1)*(e0/e1);
double sbar = GetRoot(r0 , z0 , z1 , g);
x0 = r0 * y0 /( sbar + r0 );
x1 = y1 /( sbar + 1 );
distance = Math.Sqrt( (x0-y0)*(x0-y0) + (x1-y1)*(x1-y1) );
}else{
x0 = y0;
x1 = y1;
distance = 0;
}
}
else // y0 == 0
x0 = 0 ; x1 = e1 ; distance = Math.Abs( y1 - e1 );
}else{ // y1 == 0
double numer0 = e0*y0 , denom0 = e0*e0 - e1*e1;
if ( numer0 < denom0 ){
double xde0 = numer0/denom0;
x0 = e0*xde0 ; x1 = e1*Math.Sqrt(1 - xde0*xde0 );
distance = Math.Sqrt( (x0-y0)*(x0-y0) + x1*x1 );
}else{
x0 = e0;
x1 = 0;
distance = Math.Abs( y0 - e0 );
}
}
return distance;
}
The following python code implements the equations described at "Distance from a Point to an Ellipse" and uses newton's method to find the roots and from that the closest point on the ellipse to the point.
Unfortunately, as can be seen from the example, it seems to only be accurate outside the ellipse. Within the ellipse weird things happen.
from math import sin, cos, atan2, pi, fabs
def ellipe_tan_dot(rx, ry, px, py, theta):
'''Dot product of the equation of the line formed by the point
with another point on the ellipse's boundary and the tangent of the ellipse
at that point on the boundary.
'''
return ((rx ** 2 - ry ** 2) * cos(theta) * sin(theta) -
px * rx * sin(theta) + py * ry * cos(theta))
def ellipe_tan_dot_derivative(rx, ry, px, py, theta):
'''The derivative of ellipe_tan_dot.
'''
return ((rx ** 2 - ry ** 2) * (cos(theta) ** 2 - sin(theta) ** 2) -
px * rx * cos(theta) - py * ry * sin(theta))
def estimate_distance(x, y, rx, ry, x0=0, y0=0, angle=0, error=1e-5):
'''Given a point (x, y), and an ellipse with major - minor axis (rx, ry),
its center at (x0, y0), and with a counter clockwise rotation of
`angle` degrees, will return the distance between the ellipse and the
closest point on the ellipses boundary.
'''
x -= x0
y -= y0
if angle:
# rotate the points onto an ellipse whose rx, and ry lay on the x, y
# axis
angle = -pi / 180. * angle
x, y = x * cos(angle) - y * sin(angle), x * sin(angle) + y * cos(angle)
theta = atan2(rx * y, ry * x)
while fabs(ellipe_tan_dot(rx, ry, x, y, theta)) > error:
theta -= ellipe_tan_dot(
rx, ry, x, y, theta) / \
ellipe_tan_dot_derivative(rx, ry, x, y, theta)
px, py = rx * cos(theta), ry * sin(theta)
return ((x - px) ** 2 + (y - py) ** 2) ** .5
Here's an example:
rx, ry = 12, 35 # major, minor ellipse axis
x0 = y0 = 50 # center point of the ellipse
angle = 45 # ellipse's rotation counter clockwise
sx, sy = s = 100, 100 # size of the canvas background
dist = np.zeros(s)
for x in range(sx):
for y in range(sy):
dist[x, y] = estimate_distance(x, y, rx, ry, x0, y0, angle)
plt.imshow(dist.T, extent=(0, sx, 0, sy), origin="lower")
plt.colorbar()
ax = plt.gca()
ellipse = Ellipse(xy=(x0, y0), width=2 * rx, height=2 * ry, angle=angle,
edgecolor='r', fc='None', linestyle='dashed')
ax.add_patch(ellipse)
plt.show()
Which generates an ellipse and the distance from the boundary of the ellipse as a heat map. As can be seen, at the boundary the distance is zero (deep blue).
Given an ellipse E in parametric form and a point P
the square of the distance between P and E(t) is
The minimum must satisfy
Using the trigonometric identities
and substituting
yields the following quartic equation:
Here's an example C function that solves the quartic directly and computes sin(t) and cos(t) for the nearest point on the ellipse:
void nearest(double a, double b, double x, double y, double *ecos_ret, double *esin_ret) {
double ax = fabs(a*x);
double by = fabs(b*y);
double r = b*b - a*a;
double c, d;
int switched = 0;
if (ax <= by) {
if (by == 0) {
if (r >= 0) { *ecos_ret = 1; *esin_ret = 0; }
else { *ecos_ret = 0; *esin_ret = 1; }
return;
}
c = (ax - r) / by;
d = (ax + r) / by;
} else {
c = (by + r) / ax;
d = (by - r) / ax;
switched = 1;
}
double cc = c*c;
double D0 = 12*(c*d + 1); // *-4
double D1 = 54*(d*d - cc); // *4
double D = D1*D1 + D0*D0*D0; // *16
double St;
if (D < 0) {
double t = sqrt(-D0); // *2
double phi = acos(D1 / (t*t*t));
St = 2*t*cos((1.0/3)*phi); // *2
} else {
double Q = cbrt(D1 + sqrt(D)); // *2
St = Q - D0 / Q; // *2
}
double p = 3*cc; // *-2
double SS = (1.0/3)*(p + St); // *4
double S = sqrt(SS); // *2
double q = 2*cc*c + 4*d; // *2
double l = sqrt(p - SS + q / S) - S - c; // *2
double ll = l*l; // *4
double ll4 = ll + 4; // *4
double esin = (4*l) / ll4;
double ecos = (4 - ll) / ll4;
if (switched) {
double t = esin;
esin = ecos;
ecos = t;
}
*ecos_ret = copysign(ecos, a*x);
*esin_ret = copysign(esin, b*y);
}
Try it online!
You just need to calculate the intersection of the line [P1,P0] to your elipse which is S1.
If the line equeation is:
and the elipse equesion is:
than the values of S1 will be:
Now you just need to calculate the distance between S1 to P1 , the formula (for A,B points) is:
I've solved the distance issue via focal points.
For every point on the ellipse
r1 + r2 = 2*a0
where
r1 - Euclidean distance from the given point to focal point 1
r2 - Euclidean distance from the given point to focal point 2
a0 - semimajor axis length
I can also calculate the r1 and r2 for any given point which gives me another ellipse that this point lies on that is concentric to the given ellipse. So the distance is
d = Abs((r1 + r2) / 2 - a0)
As propposed by user3235832
you shall solve quartic equation to find the normal to the ellipse (https://www.mathpages.com/home/kmath505/kmath505.htm). With good initial value only few iterations are needed (I use it myself). As an initial value I use S1 from your picture.
The fastest method I guess is
http://wwwf.imperial.ac.uk/~rn/distance2ellipse.pdf
Which has been mentioned also by Matt but as he found out the method doesn't work very well inside of ellipse.
The problem is the theta initialization.
I proposed an stable initialization:
Find the intersection of ellipse and horizontal line passing the point.
Find the other intersection using vertical line.
Choose the one that is closer the point.
Calculate the initial angle based on that point.
I got good results with no issue inside and outside:
As you can see in the following image it just iterated about 3 times to reach 1e-8. Close to axis it is 1 iteration.
The C++ code is here:
double initialAngle(double a, double b, double x, double y) {
auto abs_x = fabs(x);
auto abs_y = fabs(y);
bool isOutside = false;
if (abs_x > a || abs_y > b) isOutside = true;
double xd, yd;
if (!isOutside) {
xd = sqrt((1.0 - y * y / (b * b)) * (a * a));
if (abs_x > xd)
isOutside = true;
else {
yd = sqrt((1.0 - x * x / (a * a)) * (b * b));
if (abs_y > yd)
isOutside = true;
}
}
double t;
if (isOutside)
t = atan2(a * y, b * x); //The point is outside of ellipse
else {
//The point is inside
if (xd < yd) {
if (x < 0) xd = -xd;
t = atan2(y, xd);
}
else {
if (y < 0) yd = -yd;
t = atan2(yd, x);
}
}
return t;
}
double distanceToElipse(double a, double b, double x, double y, int maxIter = 10, double maxError = 1e-5) {
//std::cout <<"p="<< x << "," << y << std::endl;
auto a2mb2 = a * a - b * b;
double t = initialAngle(a, b, x, y);
auto ct = cos(t);
auto st = sin(t);
int i;
double err;
for (i = 0; i < maxIter; i++) {
auto f = a2mb2 * ct * st - x * a * st + y * b * ct;
auto fp = a2mb2 * (ct * ct - st * st) - x * a * ct - y * b * st;
auto t2 = t - f / fp;
err = fabs(t2 - t);
//std::cout << i + 1 << " " << err << std::endl;
t = t2;
ct = cos(t);
st = sin(t);
if (err < maxError) break;
}
auto dx = a * ct - x;
auto dy = b * st - y;
//std::cout << a * ct << "," << b * st << std::endl;
return sqrt(dx * dx + dy * dy);
}
I've recently implemented an image warping method using OpenCV 2.31(C++). However, this
method is quite time consuming... After some investigations and improvement
I succeed to reduce the processing time from 400ms to about 120ms which is quite nice.
I achieve this result by unrolling the loop (It reduces the time from 400ms to 330ms)
then I enabled optimization flags on my VC++ compiler 2008 express edition (Enabled O2 flag) - this last fix improved the processing to around 120ms.
However, since I have some other processing to implement around this warp, I'd like to reduce down this processing time even more to 20ms - lower than this value will be better of course, but I don't know if it's possible!!!...
One more thing, I'd like to do this using freely available libraries.
All suggestions are more than welcomed.
Bellow, you'll find the method I'm speaking about.
thanks for your help
Ariel B.
cv::Mat Warp::pieceWiseWarp(const cv::Mat &Isource, const cv::Mat &s, TYPE_CONVERSION type)
{
cv::Mat Idest(roi.height,roi.width,Isource.type(),cv::Scalar::all(0));
float xi, xj, xk, yi, yj, yk, x, y;
float X2X1,Y2Y1,X2X,Y2Y,XX1,YY1,X2X1_Y2Y1,a1, a2, a3, a4,b1,b2,c1,c2;
int x1, y1, x2, y2;
char k;
int nc = roi.width;
int nr = roi.height;
int channels = Isource.channels();
int N = nr * nc;
float *alphaPtr = alpha.ptr<float>(0);
float *betaPtr = beta.ptr<float>(0);
char *triMaskPtr = triMask.ptr<char>(0);
uchar *IdestPtr = Idest.data;
for(int i = 0; i < N; i++, IdestPtr += channels - 1)
if((k = triMaskPtr[i]) != -1)// the pixel do belong to delaunay
{
cv::Vec3b t = trianglesMap.row(k);
xi = s.col(1).at<float>(t[0]); yi = s.col(0).at<float>(t[0]);
xj = s.col(1).at<float>(t[1]); yj = s.col(0).at<float>(t[1]);
xk = s.col(1).at<float>(t[2]); yk = s.col(0).at<float>(t[2]);
x = xi + alphaPtr[i]*(xj - xi) + betaPtr[i]*(xk - xi);
y = yi + alphaPtr[i]*(yj - yi) + betaPtr[i]*(yk - yi);
//...some bounds checking here...
x2 = ceil(x); x1 = floor(x);
y2 = ceil(y); y1 = floor(y);
//2. use bilinear interpolation on the pixel location - see wiki for formula...
//...3. copy the resulting intensity (GL) to the destination (i,j)
X2X1 = (x2 - x1);
Y2Y1 = (y2 - y1);
X2X = (x2 - x);
Y2Y = (y2 - y);
XX1 = (x - x1);
YY1 = (y - y1);
X2X1_Y2Y1 = X2X1*Y2Y1;
a1 = (X2X*Y2Y)/(X2X1_Y2Y1);
a2 = (XX1*Y2Y)/(X2X1_Y2Y1);
a3 = (X2X*YY1)/(X2X1_Y2Y1);
a4 = (XX1*YY1)/(X2X1_Y2Y1);
b1 = (X2X/X2X1);
b2 = (XX1/X2X1);
c1 = (Y2Y/Y2Y1);
c2 = (YY1/Y2Y1);
for(int c = 0; c < channels; c++)// Consider implementing this bilinear interpolation elsewhere in another function
{
if(x1 != x2 && y1 != y2)
IdestPtr[i + c] = Isource.at<cv::Vec3b>(y1,x1)[c]*a1
+ Isource.at<cv::Vec3b>(y2,x1)[c]*a2
+ Isource.at<cv::Vec3b>(y1,x2)[c]*a3
+ Isource.at<cv::Vec3b>(y2,x2)[c]*a4;
if(x1 == x2 && y1 == y2)
IdestPtr[i + c] = Isource.at<cv::Vec3b>(y1,x1)[c];
if(x1 != x2 && y1 == y2)
IdestPtr[i + c] = Isource.at<cv::Vec3b>(y1,x1)[c]*b1 + Isource.at<cv::Vec3b>(y1,x2)[c]*b2;
if(x1 == x2 && y1 != y2)
IdestPtr[i + c] = Isource.at<cv::Vec3b>(y1,x1)[c]*c1 + Isource.at<cv::Vec3b>(y2,x1)[c]*c2;
}
}
if(type == CONVERT_TO_CV_32FC3)
Idest.convertTo(Idest,CV_32FC3);
if(type == NORMALIZE_TO_1)
Idest.convertTo(Idest,CV_32FC3,1/255.);
return Idest;
}
I would suggest:
1.Change division by a common factor to multiplication.
i.e. from a = a1/d; b = b1/d to d_1 = 1/d; a = a1*d_1; b = b1*d_1
2.Eliminate the four if test to a single bilinear interpolation.
I'm not sure whether that would help you. You could have a try.
Looking for the quickest way to calculate a point that lies on a line
a given distance away from the end point of the line:
void calculate_line_point(int x1, int y1, int x2, int y2, int distance, int *px, int *py)
{
//calculate a point on the line x1-y1 to x2-y2 that is distance from x2-y2
*px = ???
*py = ???
}
Thanks for the responses, no this is not homework, just some hacking out of
my normal area of expertise.
This is the function suggested below. It's not close to working. If I
calculate points every 5 degrees on the upper right 90 degree portion of
a circle as starting points and call the function below with the center of the circle as x2,y2 with a distance of 4 the end points are totally wrong. They lie below and to the right of the center and the length is as long as the center point. Anyone have any suggestions?
void calculate_line_point(int x1, int y1, int x2, int y2, int distance)
{
//calculate a point on the line x1-y1 to x2-y2 that is distance from x2-y2
double vx = x2 - x1; // x vector
double vy = y2 - y1; // y vector
double mag = sqrt(vx*vx + vy*vy); // length
vx /= mag;
vy /= mag;
// calculate the new vector, which is x2y2 + vxvy * (mag + distance).
px = (int) ( (double) x2 + vx * (mag + (double)distance) );
py = (int) ( (double) y2 + vy * (mag + (double)distance) );
}
I've found this solution on stackoverflow but don't understand it completely, can anyone clarify?
I think this belongs on MathOverflow, but I'll answer since this is your first post.
First you calculate the vector from x1y1 to x2y2:
float vx = x2 - x1;
float vy = y2 - y1;
Then calculate the length:
float mag = sqrt(vx*vx + vy*vy);
Normalize the vector to unit length:
vx /= mag;
vy /= mag;
Finally calculate the new vector, which is x2y2 + vxvy * (mag + distance).
*px = (int)((float)x1 + vx * (mag + distance));
*py = (int)((float)y1 + vy * (mag + distance));
You can omit some of the calculations multiplying with distance / mag instead.
These equations are wrong:
px = (int) ( (double) x2 + vx * (mag + (double)distance) );
py = (int) ( (double) y2 + vy * (mag + (double)distance) );
The correct equations are:
px = (int) ( (double) x2 + vx * (double)distance );
py = (int) ( (double) y2 + vy * (double)distance );
Tom