Geometry

Geometry studies objects that have shape, distance, angle, position, and transformation structure. In this wiki, geometry is a math foundation used by geometric deep learning, molecular modeling, protein structures, robotics-style agents, and vision.

The basic Euclidean setting represents points as vectors:

$$x\in {\mathbb{R}}^d$$

A collection of $n$ points is often written as a coordinate matrix:

$$X=\left[\begin{array}{c}{x}_1^{\top }\\ ⋮\\ x_n^{\top }\end{array}\right]\in {\mathbb{R}}^{n\times d}$$

The distance between two points is:

$$d(x,y)=\Vert x-y{\Vert }_2=\sqrt{\sum _{i=1}^d(x_i-y_i{)}^2}$$

An inner product defines angles:

$$⟨x,y⟩=x^{\top }y,\cos \theta =\frac{x^{\top }y}{\Vert x{\Vert }_2\Vert y{\Vert }_2}$$

Transformations

Many geometry-aware models ask whether a quantity should change when the coordinate frame changes. A rigid transform in 3D is:

$$x^{\prime }=Rx+t$$

where $R$ is a rotation matrix and $t$ is a translation vector.

Distances are preserved under rigid transforms:

$$\Vert (R{x}_i+t)-(R{x}_j+t){\Vert }_2=\Vert x_i-x_j{\Vert }_2$$

This is the math reason why molecular identity should not depend on the arbitrary placement of a structure in a coordinate file.

Coordinate Frame

A coordinate frame is the convention used to assign numbers to geometry. The same object can have different coordinates under a different frame.

Quantity	Frame behavior
pairwise distance $\Vert x_i-x_j{\Vert }_2$	invariant to rotation and translation
angle between vectors	invariant to rotation
coordinate $x_i$	changes with frame
displacement vector $x_i-x_j$	rotates with frame
scalar field on points	often invariant or equivariant depending on target

This distinction is central for structure models: coordinates are not arbitrary noise, but neither are they absolute labels unless the frame has physical meaning.

Distance Matrix

For point sets, pairwise distances form a matrix:

$$D_{ij}=\Vert x_i-x_j{\Vert }_2$$

The distance matrix is invariant to rigid transformations:

$$D(RX+1t^{\top })=D(X)$$

It loses orientation and chirality information, but it is often useful for contact maps, molecular geometry, and structure comparison.

Angles and Inner Products

Angles can be computed from normalized vectors:

$$\cos \theta =\frac{u^{\top }v}{\Vert u{\Vert }_2\Vert v{\Vert }_2}$$

For three points $a,b,c$, the angle at $b$ uses:

$$u=a-b,v=c-b$$

This is why geometric features often use distances, bond angles, torsion angles, local frames, or relative coordinates rather than absolute coordinates.

Key Ideas

• Points, vectors, distances, angles, frames, and transformations are the core objects.
• Some targets are geometric scalars, such as distance, energy, score, or affinity.
• Some targets are geometric objects, such as coordinates, directions, fields, or frames.
• If a task is coordinate-dependent, ask which transformations should preserve the answer.
• Geometry is the foundation; geometric deep learning is the neural modeling layer built on top.
• Distance-only representations are invariant but may discard orientation-sensitive information.
• Frame-dependent outputs must specify how they transform.

Practical Checks

• What is the geometric object: point cloud, graph with coordinates, surface, grid, curve, or manifold?
• Which quantities are invariant to coordinate transforms?
• Which quantities should rotate, translate, or reflect with the input?
• Is the coordinate frame arbitrary, experimentally meaningful, or introduced by preprocessing?
• Does the representation use absolute coordinates, relative coordinates, distances, or local frames?
• Is chirality or orientation important for the task?

Related

• Linear algebra
• Symmetry group
• Vector norm and similarity
• Geometric deep learning
• Structure
• Protein-ligand complex