I first ran into tangent space while learning about normal mapping. It was described as this in-between space that connects surfaces and UVs, something you need to make lighting work. Nobody really explained what it was. Tutorials showed math and shader code snippets, but none of them answered the real question: what is tangent space and why does it exist at all?
When I first learned about normal mapping I didn’t care as long as the normals looked right. But while working on mesh processing for VGLX that answer stopped being enough. I wanted to understand what those tangent vectors meant, not just how to compute them. What geometry were they pointing to? What was this “space” actually describing?
Eventually I realized the answer wasn’t mysterious at all. Tangent space isn’t a rendering trick. It’s a geometric structure that appears any time a surface has a parameterization. I just hadn’t connected the dots before.
When we define tangent space using UV coordinates, it becomes the bridge between the flat world of texture coordinates and the curved world of 3D surfaces. Once you see that connection, normal mapping suddenly makes perfect sense.
In this article I’ll take tangent space apart piece by piece: what it really is, how it emerges from UVs, how it’s computed, and how it forms the foundation for normal mapping and other techniques that depend on local surface orientation.
Tangent space isn’t a global coordinate system. It’s a local frame built independently at every point on a surface. Each point has its own small world, a tiny patch of geometry defined by the surface itself. The tangent plane is that world’s foundation: a flat approximation that captures how the surface behaves locally and where the space gets its name.
Every point on a smooth surface has a tangent plane defined by its normal vector. We can see the same idea in two dimensions: to find the normal at a point on a curve, we first take its tangent line, then rotate it ninety degrees. The tangent plane is the 3D version of that relationship.
Tangent plane at a point on the surface with two tangent vectors lying within the plane.
Vectors that lie on this plane are called tangent vectors. We can pick two perpendicular directions within the plane that together with the normal form an orthonormal basis: a local coordinate frame on the surface that we call tangent space.
Tangent space lets us express directions, derivatives, and transformations relative to the surface itself rather than the world. That's important because most shading computations depend on directions defined locally: light hitting a surface, a normal perturbation from a texture, or the direction of anisotropy.
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