Modern bump mapping (also known as normal mapping) requires that tangent plane basis vectors be calculated for each vertex in a mesh. This article presents the theory behind the computation of per-vertex tangent spaces for an arbitrary triangle mesh and provides source code that implements the proper mathematics.

[This derivation also appears in
*Mathematics for 3D Game Programming and Computer Graphics, 3rd ed.*, Section 7.8.]

We want our tangent space to be aligned such that the *x* axis corresponds to the *u* direction in the bump map and
the *y* axis corresponds to the *v* direction in the bump map. That is, if **Q** represents a point inside the triangle,
we would like to be able to write

where **P**_{0} is the position of one of the vertices of the triangle, and (*u*_{0}, *v*_{0})
are the texture coordinates at that vertex. The vectors **T** and **B** are the *tangent* and *bitangent* vectors aligned to
the texture map, and these are what we’d like to calculate.

Suppose that we have a triangle whose vertex positions are given by the points **P**_{0}, **P**_{1}, and
**P**_{2}, and whose corresponding texture coordinates are given by (*u*_{0}, *v*_{0}),
(*u*_{1}, *v*_{1}), and (*u*_{2}, *v*_{2}). Our calculations can be
made much simpler by working relative to the vertex **P**_{0}, so we let

and

(*s*_{1}, *t*_{1}) = (*u*_{1} − *u*_{0}, *v*_{1} − *v*_{0})

(*s*_{2}, *t*_{2}) = (*u*_{2} − *u*_{0}, *v*_{2} − *v*_{0}).

(

We need to solve the following equations for **T** and **B**.

This is a linear system with six unknowns (three for each **T** and **B**) and six equations (the *x*, *y*, and *z*
components of the two vector equations). We can write this in matrix form as follows.

(Q_{1})_{x}( Q_{2})_{x} |
(Q_{1})_{y}( Q_{2})_{y} |
(Q_{1})_{z}( Q_{2})_{z} |
= | s_{1}s_{2} |
t_{1}t_{2} |
T_{x}B_{x} |
T_{y}B_{y} |
T_{z}B_{z} |

Multiplying both sides by the inverse of the (*s*, *t*) matrix, we have

T_{x}B_{x} |
T_{y}B_{y} |
T_{z}B_{z} |
= | 1s_{1}t_{2} − s_{2}t_{1} |
t_{2}−s_{2} |
−t_{1}s_{1} |
(Q_{1})_{x}( Q_{2})_{x} |
(Q_{1})_{y}( Q_{2})_{y} |
(Q_{1})_{z}( Q_{2})_{z} |
. |

This gives us the (unnormalized) **T** and **B** vectors for the triangle whose vertices are **P**_{0},
**P**_{1}, and **P**_{2}. To find the tangent vectors for a single vertex, we average the tangents for all
triangles sharing that vertex in a manner similar to the way in which vertex normals are commonly calculated. In the case that
neighboring triangles have discontinuous texture mapping, vertices along the border are generally already duplicated since they
have different mapping coordinates anyway. We do not average tangents from such triangles because the result would not accurately
represent the orientation of the bump map for either triangle.

Once we have the normal vector **N** and the tangent vectors **T** and **B** for a vertex, we can transform from tangent
space into object space using the matrix

T_{x}T_{y}T_{z} |
B_{x}B_{y}B_{z} |
N_{x}N_{y}N_{z} |
. |

To transform in the opposite direction (from object space to tangent space—what we want to do to the light direction), we can
simply use the inverse of this matrix. It is not necessarily true that the tangent vectors are perpendicular to each other or to the
normal vector, so the inverse of this matrix is not generally equal to its transpose. It is safe to assume, however, that the three
vectors will at least be close to orthogonal, so using the Gram-Schmidt algorithm to orthogonalize them should not cause any unacceptable
distortions. Using this process, new (still unnormalized) tangent vectors **T′** and **B′** are given by

Normalizing these vectors and storing them as the tangent and bitangent for a vertex lets us use the matrix

T′_{x}B′_{x}N_{x} |
T′_{y}B′_{y}N_{y} |
T′_{z}B′_{z}N_{z} |
(*) |

to transform the direction to light from object space into tangent space. Taking the dot product of the transformed light direction with a sample from the bump map then produces the correct Lambertian diffuse lighting value.

It is not necessary to store an extra array containing the per-vertex bitangent since the cross product **N** × **T′**
can be used to obtain *m***B′**, where *m* = ±1 represents the handedness of the tangent space.
The handedness value must be stored per-vertex since the bitangent **B′** obtained from **N** × **T′**
may point in the wrong direction. The value of *m* is equal to the determinant of the matrix in Equation (*). You might find it convenient
to store the per-vertex tangent vector **T′** as a four-dimensional entity whose *w* coordinate holds the value of *m*.
Then the bitangent **B′** can be computed using the formula

where the cross product ignores the *w* coordinate. This works nicely for vertex shaders by avoiding the need to specify an
additional array containing the per-vertex *m* values.

The term *binormal* is commonly used as the name of the second tangent direction
(that is perpendicular to the surface normal and *u*-aligned tangent direction). This is a misnomer.
The term binormal pops up in the study of **curves** and completes what is known as a Frenet frame about
a particular point on a curve. Curves have a single tangent direction and two orthogonal normal directions,
hence the terms normal and binormal. When discussing a coordinate frame at a point on a **surface**,
there is one normal direction and two tangent directions, which should be called the tangent and *bitangent*.

The code below generates a four-component tangent **T** in which the handedness of the local coordinate
system is stored as ±1 in the *w*-coordinate. The bitangent vector **B** is
then given by **B** = (**N** × **T**) · *T _{w}*.

#include "Vector4D.h" struct Triangle { unsigned short index[3]; }; void CalculateTangentArray(long vertexCount, const Point3D *vertex, const Vector3D *normal, const Point2D *texcoord, long triangleCount, const Triangle *triangle, Vector4D *tangent) { Vector3D *tan1 = new Vector3D[vertexCount * 2]; Vector3D *tan2 = tan1 + vertexCount; ZeroMemory(tan1, vertexCount * sizeof(Vector3D) * 2); for (long a = 0; a < triangleCount; a++) { long i1 = triangle->index[0]; long i2 = triangle->index[1]; long i3 = triangle->index[2]; const Point3D& v1 = vertex[i1]; const Point3D& v2 = vertex[i2]; const Point3D& v3 = vertex[i3]; const Point2D& w1 = texcoord[i1]; const Point2D& w2 = texcoord[i2]; const Point2D& w3 = texcoord[i3]; float x1 = v2.x - v1.x; float x2 = v3.x - v1.x; float y1 = v2.y - v1.y; float y2 = v3.y - v1.y; float z1 = v2.z - v1.z; float z2 = v3.z - v1.z; float s1 = w2.x - w1.x; float s2 = w3.x - w1.x; float t1 = w2.y - w1.y; float t2 = w3.y - w1.y; float r = 1.0F / (s1 * t2 - s2 * t1); Vector3D sdir((t2 * x1 - t1 * x2) * r, (t2 * y1 - t1 * y2) * r, (t2 * z1 - t1 * z2) * r); Vector3D tdir((s1 * x2 - s2 * x1) * r, (s1 * y2 - s2 * y1) * r, (s1 * z2 - s2 * z1) * r); tan1[i1] += sdir; tan1[i2] += sdir; tan1[i3] += sdir; tan2[i1] += tdir; tan2[i2] += tdir; tan2[i3] += tdir; triangle++; } for (long a = 0; a < vertexCount; a++) { const Vector3D& n = normal[a]; const Vector3D& t = tan1[a]; // Gram-Schmidt orthogonalize tangent[a] = (t - n * Dot(n, t)).Normalize(); // Calculate handedness tangent[a].w = (Dot(Cross(n, t), tan2[a]) < 0.0F) ? -1.0F : 1.0F; } delete[] tan1; }

Lengyel, Eric. “Computing Tangent Space Basis Vectors for an Arbitrary Mesh”. Terathon Software, 2001. http://terathon.com/code/tangent.html

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