Mapping a dot on a 2D image of a ball to its true 3D position on the surface of the ball

Note: True dot mapping is required to determin the amount of both back and side spin rotation

AI says:

"To map a dot on a 2D image of a spherical ball to its true 3D position on the surface of that ball, you must correct for perspective distortion.

Because a ball is a 3D sphere, a dot near the edge of the image appears squished and closer to the center than it actually is.

To solve this, you need to calculate the dot's true 3D spherical coordinates (latitude and longitude) relative to the center of the ball."

Here is how to implement this calculation using OpenCV in C++.

This code takes the detected center of the ball, its radius, and the pixel coordinates of the dot, then outputs its true 3D surface vector.

AI generated code example for "Dot" mapping :

// Function to calculate true 3D surface position from 2D pixel coordinates
Point3D getTrue3DPosition(cv::Point2f ballCenter, double ballRadius, cv::Point2f dotPixel) {
// 1. Shift origin to the center of the ball
double x_rel = dotPixel.x - ballCenter.x;
double y_rel = ballCenter.y - dotPixel.y; // Invert Y because image pixels go down, Cartesian goes up

// 2. Calculate the 2D pixel distance from the center
double r = std::sqrt(x_rel * x_rel + y_rel * y_rel);

// Handle edge case: if the dot is mathematically outside the radius due to pixel blur
if (r > ballRadius) {
r = ballRadius;
}

// 3. Calculate the true Z depth (facing the camera)
double z_rel = std::sqrt((ballRadius * ballRadius) - (r * r));

// 4. Normalize the vector to get unit coordinates on the sphere's surface (-1.0 to 1.0)
Point3D truePos;
truePos.x = x_rel / ballRadius;
truePos.y = y_rel / ballRadius;
truePos.z = z_rel / ballRadius;

return truePos;
}

int main() {
// Example: A ball centered at (300, 300) with a 100-pixel radius
cv::Point2f center(300.0f, 300.0f);
double radius = 100.0;

// A dot detected at pixel (350, 250)
cv::Point2f dot(350.0f, 250.0f);

Point3D surfacePos = getTrue3DPosition(center, radius, dot);

std::cout << "True 3D Vector on Sphere Surface:\n";
std::cout << "X: " << surfacePos.x << " (Right)\n";
std::cout << "Y: " << surfacePos.y << " (Up)\n";
std::cout << "Z: " << surfacePos.z << " (Depth toward camera)\n";

// Convert to Latitude / Longitude angles if needed
double longitude = std::atan2(surfacePos.x, surfacePos.z) * 180.0 / CV_PI;
double latitude = std::asin(surfacePos.y) * 180.0 / CV_PI;
std::cout << "\nSpherical Coordinates:\nLon: " << longitude << "°, Lat: " << latitude << "°\n";

return 0;
}