I've seen the term Normal being used in many places, but I'm not quite sure what it is or what it's used for; what exactly are they?
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read: http://en.wikipedia.org/wiki/Normal_%28geometry%29 and http://en.wikibooks.org/wiki/Blender_3D:_Noob_to_Pro/Normals_and_Shading – Dec 20 '14 at 18:46
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Already did. That's mostly were the answer originated from. – someonewithpc Dec 20 '14 at 18:50
2 Answers
Normals are vectors that are Orthogonal to the Tangent plane of a curved surface; Or simply orthogonal to a face.
The Normals are usually used to determine a surface's orientation toward a light source for shading. Most common examples:
Flat, Goraud, Phong shading

Fresnel, Refraction, Reflection effects
- Bump, Normal, Displacement mapping
Normals are generally normalized to length 1 (in arbitrary units), otherwise calculation errors, and thus artifacts, may arise. Since they only point one way they are also used for back visibility culling (when double-sided is off).
That's why it is important for the normals to be consistent and coherent (i.e. point in the correct way - outwards of a model).
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1Normals are also used for bump and (duh) normal mapping and smooth shading. – PGmath Dec 20 '14 at 18:42
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1@NoviceInDisguise Yea, it felt kind of dumb saying that normal maps mess with normals. But then again, when I first saw a normal map I thought “that doesn't look very normal to me!” – PGmath Dec 20 '14 at 20:35
Besides the mathematical explanations, I think a picture will help to understand what is meant by normals and perpendicular. The faces having a normal (blue lines) pointing to a light source are rendered brighter than the faces pointing into other directions.

You can toggle the display of normals in the Properties Panel (right panel N):

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1+1 if you also add screenshot showing how you turn on the normal lines display. – Neil Dec 22 '14 at 08:38