diff --git a/src/05/ray.ts b/src/05/ray.ts
index 3bebd27..ad9c521 100644
--- a/src/05/ray.ts
+++ b/src/05/ray.ts
@@ -33,13 +33,17 @@ export default class Ray {
         // TODO: on the image plane. In addition to the coordinates (x, y), you will need the
         // TODO: width and height of the camera (i.e. the width and height of the camera's
         // TODO: image plane), and the angle alpha specifying the camera's field of view.
-        var temp = new Vector(0,0,0,0);
-        temp.x = x - ((camera.width-1) / 2);
-        temp.y = ((camera.height-1) / 2) - y;
-        temp.z = - ((camera.width/2) / (Math.tan(camera.alpha/2)));
+        var a = x - ((camera.width-1) / 2);
+        var b = ((camera.height-1) / 2) - y;
+        var c = - ((camera.width/2) / (Math.tan(camera.alpha/2)));
 
-        temp = temp.normalize();
+        //temp = temp.normalize();
 
-        return new Ray(camera.origin, temp);
+        var nom = Math.sqrt(a*a + b*b + c*c);
+        let direction = new Vector((x-(camera.width-1)/2)/nom,
+                        ((camera.height-1)/2-y)/nom,
+                        (-((camera.width/2)/Math.tan(camera.alpha/2)))/nom,0);
+
+        return new Ray(camera.origin, direction);
     }
 }
diff --git a/src/05/sphere.ts b/src/05/sphere.ts
index 2024b7a..2142f6a 100644
--- a/src/05/sphere.ts
+++ b/src/05/sphere.ts
@@ -45,23 +45,20 @@ export default class Sphere {
         // TODO: of two hits, return the one closer to the start point of
         // TODO: the ray.
 
-        let c = Math.pow(ray.origin.dot(ray.direction),2) - ray.origin.dot(ray.origin) + Math.pow(this.radius,2);
-        // hier oben noch nicht berechenenm da c negativ sein kann und dann wurzel nicht funktioniert
+        let newCenter = ray.origin.sub(this.center);
+        // hier oben noch nicht t berechenenm da c negativ sein kann und dann wurzel nicht funktioniert
+        let c = Math.pow(newCenter.dot(ray.direction),2) - newCenter.dot(newCenter) + Math.pow(this.radius,2);
         
         if(c < 0) {
             return null;
         } else if(c == 0) {
-            return new Intersection(t1, ); //schnittpunkt ist origin + direction * t
+            //PQ Formel
+            let t1 = - newCenter.dot(ray.direction);
+            return new Intersection(t1, newCenter.add(ray.direction.mul(t1)), (newCenter.add(ray.direction.mul(t1)).sub(this.center)).normalize() ); //schnittpunkt ist origin + direction * t
 
         } else {
-            let t1 = - ray.origin.dot(ray.direction) + Math.sqrt( Math.pow(ray.origin.dot(ray.direction),2) - ray.origin.dot(ray.origin) + Math.pow(this.radius,2));
-            let t2 = - ray.origin.dot(ray.direction) - Math.sqrt( Math.pow(ray.origin.dot(ray.direction),2) - ray.origin.dot(ray.origin) + Math.pow(this.radius,2));
-
-            if(t1 < t2) {
-            return null;
-            } else {
-            return null;
-            }
+            let t2 = - newCenter.dot(ray.direction) - Math.sqrt(c);
+            return new Intersection(t2, newCenter.add(ray.direction.mul(t2)), (newCenter.add(ray.direction.mul(t2)).sub(this.center)).normalize() ); //schnittpunkt ist origin + direction * t
         }
     }
 }