Chapter 6: Accuracy issue with transformed sphere normals

First, thanks Jamis for the awesome book! Having a blast going through it.

Thought I'd raise this here regarding the sphere normal tests on Page 80 of the paper copy, section "Transforming Normals".

The values quoted in the tests are not very precise and my tests failed to pass.

Specifically, the tests as specified

Scenario: Computing the normal on a translated sphere
 Given s ← sphere()
   And set_transform(s, translation(0, 1, 0))
 When n ← normal_at(s, point(0, 1.70711, -0.70711))
 Then n = vector(0, 0.70711, -0.70711)

Scenario: Computing the normal on a transformed sphere
 Given s ← sphere()
   And m ← scaling(1, 0.5, 1) * rotation_z(π/5)
   And set_transform(s, m)
 When n ← normal_at(s, point(0, √2/2, -√2/2))
 Then n = vector(0, 0.97014, -0.24254)


In particular, the first test is not a point on the sphere. (quick check: √ (0.70711^2 + 0.70711^2) != 1.0

It does work within the book's tolerance of 0.00001, but I'm using a stricter comparison and my tests failed. For others who come across this issue, I recommend changing the tests to the following:

Scenario: Computing the normal on a translated sphere
 Given s ← sphere()
   And set_transform(s, translation(0, 1, 0))
 When n ← normal_at(s, point(0, 1+√2/2, -√2/2))
 Then n = vector(0, 1/√2, -1/√2)

Scenario: Computing the normal on a transformed sphere
 Given s ← sphere()
   And m ← scaling(1, 0.5, 1) * rotation_z(π/5)
   And set_transform(s, m)
 When n ← normal_at(s, point(0, √2/2, -√2/2))
 Then n = vector(0, 0.970142500,-0.242535625)


Which worked for me.

A quick check can show the difference in accuracy:

>>> x,y = 0.970142500,-0.242535625
>>> x*x+y*y
0.9999999997003907
>>> a,b = .97014, -0.24254
>>> a*a+b*b
0.9999972711999999