# Intersection of a vector with a sphere

### Triv Prasad ・4 min read

A good friend of mine wanted some help with a VR project to see if a ray (controller's motion vector) is intersecting with a spherical viewport of the user. Here's a rudimentary java program to check for a vector intersecting a sphere.

User input <- Coordinates of the two points of the vector and the coordinates of the sphere center and sphere radius

Output -> Coordinates of the point(s) of intersection

Have not programmed in Java for a couple of decades so was an interesting and enriching experience as the program slowly got built out, refined and refactored (to the best of my ability). Sure it's not computationally efficient but hoping to make it an API to use within Unity at some point!

```
/** Copyright - Trivikram Prasad
Date: May 19, 2018
All Rights Reserved
*/
// Program to calculate the points of intersection of a vector with a Sphere
import java.util.Scanner;
// DEFINE ALL CLASSES
// 3D Point class
class Point3D{
double x,y,z; //Coordinates of a point
public Point3D (double x, double y, double z)
{
this.x = x;
this.y = y;
this.z = z;
}
}
// Sphere Class
class Sphere{
double cX, cY, cZ; // Coordinates of the sphere center
double cR; // Radius of sphere
public Sphere (double cX, double cY, double cZ, double cR)
{
this.cX = cX;
this.cY = cY;
this.cZ = cZ;
this.cR = cR;
}
}
// String class -- only for formatting output
class StringPoint{
String pX, pY, pZ; // Converting coordinate values to string values for formatting
public StringPoint (String pX, String pY, String pZ)
{
this.pX = pX;
this.pY = pY;
this.pZ = pZ;
}
}
// The main class. This is where the real s**t happens
public class lineSphereX {
public static void main(String[] args){
double a,b,c; // Coefficients of the quadratic equation
double discr; // Square of the discriminant of the quadratic
double t1 = 0.0; // parameter 1
double t2 = 0.0; // parameter 2
Scanner in = new Scanner(System.in);
// Read coordinates of the first point on the line
System.out.print("Enter x, y, z coordinates of the 1st point on the line: ");
Point3D point1 = readLineInputCoordinates(in);
// Read coordinates of the second point on the line
System.out.print("Enter x, y, z coordinates of the 2nd point on the line: ");
Point3D point2 = readLineInputCoordinates(in);
// Read center coordinates and radius of sphere
System.out.print("Enter x, y, z center coordinates and RADIUS 'r' of the sphere: ");
Sphere sphCenter= readSphereInputCoordinates(in);
in.close();
//Calculate coefficients: 'a', 'b', 'c' for the equation 'ax^2 + bx + c = 0
a = Math.pow(point2.x - point1.x,2) + Math.pow(point2.y - point1.y,2) + Math.pow(point2.z - point1.z,2);
b = 2 * ((point2.x-point1.x) * (point1.x-sphCenter.cX) + (point2.y-point1.y) * (point1.y-sphCenter.cY)+ (point2.z-point1.z) * (point1.z-sphCenter.cZ));
c = Math.pow((point1.x - sphCenter.cX),2) + Math.pow((point1.y - sphCenter.cY),2) + Math.pow((point1.z - sphCenter.cZ),2) - Math.pow(sphCenter.cR, 2);
discr = Math.pow(b, 2) - (4*a*c);
// No intersection points`
if (discr < 0)
{
System.out.println("\nVector does not intersect the sphere");
return;
}
else if(discr == 0) // Single intersection point
{
System.out.println("\nVector intersects the sphere at a single point:");
if (a > 0)
{
t1 = -b/(2*a);
StringPoint XPoint = findXPoint(point1, point2, t1);
System.out.println("The tangent to sphere is at " + XPoint.pX + ","+ XPoint.pY + "," + XPoint.pZ + "\n");
}
else{
System.out.println("Not a line\n");
}
}
else // Two intersection points
{
System.out.println("\nVector intersects the sphere at two points");
t1 = (-b - Math.sqrt(discr))/(2*a); // 1st Vector equation parameter
t2 = (-b + Math.sqrt(discr))/(2*a); // 2nd Vector equation parameter
// Find the points of intersection
StringPoint XPoint1 = findXPoint(point1, point2, t1);
StringPoint XPoint2 = findXPoint(point1, point2, t2);
System.out.println("1st point of vector intersection with sphere: " + XPoint1.pX + "," + XPoint1.pY + "," + XPoint1.pZ);
System.out.println("2nd point of vector intersection with sphere: " + XPoint2.pX + "," + XPoint2.pY + "," + XPoint2.pZ + "\n");
}
System.exit(0);
}
// Function for user input of sphere center coordinates and radius
private static Sphere readSphereInputCoordinates(Scanner input) {
double sphere_ctr [] = new double[4];
Sphere input_sphere = new Sphere(0,0,0,0);
for (int i = 0; i < 4; i++)
{
sphere_ctr[i] = input.nextDouble();
}
input_sphere.cX= sphere_ctr[0];
input_sphere.cY = sphere_ctr[1];
input_sphere.cZ = sphere_ctr[2];
input_sphere.cR = sphere_ctr[3];
return input_sphere;
}
// Function for user inputs of point coordinates
private static Point3D readLineInputCoordinates(Scanner input) {
double line_pt [] = new double[3];
Point3D input_coords = new Point3D(0,0,0);
for (int i = 0; i < 3; i++)
{
line_pt[i] = input.nextDouble();
}
input_coords.x = line_pt[0];
input_coords.y = line_pt[1];
input_coords.z = line_pt[2];
return input_coords;
}
// Function to calculate the point of intersection
private static StringPoint findXPoint(Point3D point1, Point3D point2, double t) {
Point3D intPoint = new Point3D(0,0,0);
StringPoint XStrPoint = new StringPoint("","","");
// Parametric equation of the form L = P + tU
// where 'L' is the intersection point, 'P' is the point on the line and
// U is the unit vector (Point2 - Point1)
intPoint.x = point1.x + t * (point2.x - point1.x);
intPoint.y = point1.y + t * (point2.y - point1.y);
intPoint.z = point1.z + t * (point2.z - point1.z);
XStrPoint = formatString(intPoint);
return XStrPoint;
}
// Function to format the coordinates to string for display/output to console
private static StringPoint formatString(Point3D intPoint) {
StringPoint strPoint = new StringPoint("","","");
strPoint.pX = String.format("%.2f", intPoint.x);
strPoint.pY= String.format("%.2f", intPoint.y);
strPoint.pZ = String.format("%.2f", intPoint.z);
return strPoint;
}
}
```