Geometry and structural losses

Express Points in Local Frames

EasyRaw PyTorchAbout 10 min2 public examples plus private tests

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Your task

Express every world-space point in every local coordinate frame. Return a tensor whose frame and point axes enumerate all frame-point combinations.

Function signature

points_in_local_frames(
    rotation: torch.Tensor,
    translation: torch.Tensor,
    points: torch.Tensor,
) -> torch.Tensor

Worked example

One point, two coordinate frames

The same point has local coordinates (1, 0, 0) in the identity frame and (0, -1, 0) in the rotated frame.

import torch

rotation = torch.tensor([[[[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[0.0, -1.0, 0.0], [1.0, 0.0, 0.0], [0.0, 0.0, 1.0]]]], dtype=torch.float64, requires_grad=True)
translation = torch.tensor([[[0.0, 0.0, 0.0], [0.0, 0.0, 0.0]]], dtype=torch.float64, requires_grad=True)
points = torch.tensor([[[1.0, 0.0, 0.0]]], dtype=torch.float64, requires_grad=True)

result = points_in_local_frames(rotation=rotation, translation=translation, points=points)

Expected return

torch.tensor([[[[1.0, 0.0, 0.0]], [[0.0, -1.0, 0.0]]]], dtype=torch.float64)

Parameters

Parameter	Shape and meaning
`rotation`torch.Tensor	$(B,F,3,3)$ Local-to-world rotation for each frame; its columns are the local basis vectors in world coordinates.
`translation`torch.Tensor	$(B,F,3)$ World-space origin of each frame.
`points`torch.Tensor	$(B,P,3)$ World-space point coordinates, shared across frames within the same batch item.

Return

torch.Tensor · $(B,F,P,3)$

Return Point coordinates in every local frame, in batch/frame/point/coordinate order.

Required behavior

rotation maps local column vectors into world coordinates; translation is the frame origin in world coordinates.
For every batch item, frame, and point, compute $p_{local}=R^\top(p_{world}-t)$ . Subtract the frame origin before applying the transposed rotation.
Input shapes are rotation $(B,F,3,3)$ , translation $(B,F,3)$ , and points $(B,P,3)$ . Return local_points with shape $(B,F,P,3)$ .
Batch size is positive. F or P may be zero; preserve those empty dimensions. Inputs may be strided.
Gradients are checked for points, translations, and the nine rotation matrix entries. Use the supplied matrix directly; do not re-orthogonalize it.

Why this matters

Each residue frame supplies its own coordinate system. Comparing local coordinates measures structural agreement without first choosing one global protein alignment; the numbers of frames and points may differ.

Starter code

You write the body of this function in the BioTorch workspace. Your code runs in an isolated PyTorch sandbox; Run checks the public examples and Submit grades the full test suite, including gradients.

import torch


def points_in_local_frames(
    rotation: torch.Tensor,  # (B,F,3,3)
    translation: torch.Tensor,  # (B,F,3)
    points: torch.Tensor,  # (B,P,3)
) -> torch.Tensor:
    """Return a tensor with shape (B,F,P,3)."""
    # Implement the operation. Inputs and test setup are supplied by BioTorch.
    raise NotImplementedError("Implement points_in_local_frames")

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3 progressive hints, a solution walkthrough, and grading that checks values, shapes and gradients are waiting in the workspace.

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Sources and further reading

A primitive exercise, not an end-to-end reproduction of the named models. No affiliation or endorsement is implied.