diff --git a/ray-tracer/src/bin/projectile.rs b/ray-tracer/src/bin/projectile.rs
index 8f71c28..f71553d 100644
--- a/ray-tracer/src/bin/projectile.rs
+++ b/ray-tracer/src/bin/projectile.rs
@@ -46,7 +46,7 @@ fn main() {
         }
     }
 
-    let ppm = PPM::from(canvas);
+    let ppm = PPM::from(&canvas);
 
     let mut file = File::create("projectile.ppm").unwrap();
     let _ = file.write(ppm.as_bytes());
diff --git a/ray-tracer/src/ppm.rs b/ray-tracer/src/ppm.rs
index 575d43a..b1e1b71 100644
--- a/ray-tracer/src/ppm.rs
+++ b/ray-tracer/src/ppm.rs
@@ -91,7 +91,7 @@ mod tests {
                         0 0 0 0 0 0 0 128 0 0 0 0 0 0 0\n\
                         0 0 0 0 0 0 0 0 0 0 0 0 0 0 255\n";
 
-        let ppm = PPM::from(c);
+        let ppm = PPM::from(&c);
         assert_eq!(*ppm, expected);
     }
 
@@ -112,7 +112,7 @@ mod tests {
      	                255 204 153 255 204 153 255 204 153 255 204 153 255 204 153 255 204\n\
      	                153 255 204 153 255 204 153 255 204 153 255 204 153\n";
 
-        let ppm = PPM::from(c);
+        let ppm = PPM::from(&c);
         assert_eq!(*ppm, expected);
     }
 }
diff --git a/ray-tracer/src/types/matrix.rs b/ray-tracer/src/types/matrix.rs
index 226a42c..0e9bc36 100644
--- a/ray-tracer/src/types/matrix.rs
+++ b/ray-tracer/src/types/matrix.rs
@@ -235,10 +235,25 @@ impl std::ops::Mul<Tuple> for Matrix {
     }
 }
 
+impl std::ops::Mul<&Point> for &Matrix {
+    type Output = Point;
+    fn mul(self, rside: &Point) -> Point {
+        let t: Tuple = **rside;
+        Point::from(self * t)
+    }
+}
+
 impl std::ops::Mul<Point> for &Matrix {
     type Output = Point;
     fn mul(self, rside: Point) -> Point {
-        Point::from(self * *rside)
+        self * &rside
+    }
+}
+
+impl std::ops::Mul<&Point> for Matrix {
+    type Output = Point;
+    fn mul(self, rside: &Point) -> Point {
+        &self * rside
     }
 }
 
diff --git a/ray-tracer/src/types/point.rs b/ray-tracer/src/types/point.rs
index c8a0d98..cc8b5d2 100644
--- a/ray-tracer/src/types/point.rs
+++ b/ray-tracer/src/types/point.rs
@@ -51,10 +51,17 @@ impl std::ops::Add<Vector> for Point {
     }
 }
 
-impl std::ops::Sub for Point {
+impl std::ops::Sub<&Point> for &Point {
+    type Output = Vector;
+    fn sub(self, r: &Point) -> Self::Output {
+        Vector::from(self.0 - r.0)
+    }
+}
+
+impl std::ops::Sub<Point> for Point {
     type Output = Vector;
     fn sub(self, r: Point) -> Self::Output {
-        Vector::from(self.0 - r.0)
+        &self - &r
     }
 }
 
diff --git a/ray-tracer/src/types/sphere.rs b/ray-tracer/src/types/sphere.rs
index 07b1afe..3e9dc44 100644
--- a/ray-tracer/src/types/sphere.rs
+++ b/ray-tracer/src/types/sphere.rs
@@ -1,4 +1,4 @@
-use crate::types::{Matrix, Point};
+use crate::types::{Matrix, Point, Vector};
 
 #[derive(Debug, PartialEq)]
 pub struct Sphere {
@@ -14,6 +14,15 @@ impl Sphere {
     pub fn set_transformation(&mut self, m: Matrix) {
         self.transformation = m
     }
+
+    pub fn normal_at(&self, world_point: &Point) -> Vector {
+        let inverted_transform = self.transformation.inverse();
+        let object_point = &inverted_transform * world_point;
+        let object_normal = object_point - Point::new(0., 0., 0.);
+        let mut world_normal = inverted_transform.transpose() * *object_normal;
+        world_normal.3 = 0.;
+        Vector::from(world_normal).normalize()
+    }
 }
 
 impl Default for Sphere {
@@ -28,7 +37,7 @@ impl Default for Sphere {
 #[cfg(test)]
 mod test {
     use super::*;
-    use crate::transforms::translation;
+    use crate::transforms::{rotation_z, scaling, translation};
 
     #[test]
     fn sphere_has_default_transformation() {
@@ -43,4 +52,55 @@ mod test {
         s.set_transformation(t.clone());
         assert_eq!(*s.transformation(), t);
     }
+
+    #[test]
+    fn normal_of_sphere_on_x() {
+        let s = Sphere::default();
+        let n = s.normal_at(&Point::new(1., 0., 0.));
+        assert_eq!(n, Vector::new(1., 0., 0.));
+    }
+
+    #[test]
+    fn normal_of_sphere_on_y() {
+        let s = Sphere::default();
+        let n = s.normal_at(&Point::new(0., 1., 0.));
+        assert_eq!(n, Vector::new(0., 1., 0.));
+    }
+
+    #[test]
+    fn normal_of_sphere_on_z() {
+        let s = Sphere::default();
+        let n = s.normal_at(&Point::new(0., 0., 1.));
+        assert_eq!(n, Vector::new(0., 0., 1.));
+    }
+
+    #[test]
+    fn normal_of_sphere_on_nonaxial() {
+        let s = Sphere::default();
+        let n = s.normal_at(&Point::new(3_f64.sqrt() / 3., 3_f64.sqrt() / 3., 3_f64.sqrt() / 3.));
+        assert_eq!(n, Vector::new(3_f64.sqrt() / 3., 3_f64.sqrt() / 3., 3_f64.sqrt() / 3.));
+    }
+
+    #[test]
+    fn normal_is_normalized() {
+        let s = Sphere::default();
+        let n = s.normal_at(&Point::new(3_f64.sqrt() / 3., 3_f64.sqrt() / 3., 3_f64.sqrt() / 3.));
+        assert_eq!(n.normalize(), n);
+    }
+
+    #[test]
+    fn compute_normal_of_translated_sphere() {
+        let mut s = Sphere::default();
+        s.set_transformation(translation(0., 1., 0.));
+        let n = s.normal_at(&Point::new(0., 1.70711, -0.70711));
+        assert_eq!(n, Vector::new(0., 0.70711, -0.70711));
+    }
+
+    #[test]
+    fn compute_normal_of_transformed_sphere() {
+        let mut s = Sphere::default();
+        s.set_transformation(scaling(1., 0.5, 1.) * rotation_z(std::f64::consts::PI / 5.));
+        let n = s.normal_at(&Point::new(0., 2_f64.sqrt() / 2., -2_f64.sqrt() / 2.));
+        assert_eq!(n, Vector::new(0., 0.97014, -0.24254));
+    }
 }
diff --git a/ray-tracer/src/types/tuple.rs b/ray-tracer/src/types/tuple.rs
index 1082d14..af394e7 100644
--- a/ray-tracer/src/types/tuple.rs
+++ b/ray-tracer/src/types/tuple.rs
@@ -108,4 +108,3 @@ impl std::ops::Div<f64> for Tuple {
         &self / scalar
     }
 }
-