Explain Geometric Modelling Commands: Scaling, Mirror, and Rotation

Geometric modelling commands like Scaling , Mirror , and Rotation are typically implemented as transformations in 2D/3D. Conceptually, each command takes a set of points (the model’s vertices, edges, or faces) and maps every point to a new location using a transformation rule (often expressed with matrices). This section explains what each command does, the geometric parameters you must choose, and common modelling pitfalls.

To connect the “button clicks” to geometry, note that all three operations preserve some structure but not all: scaling preserves collinearity but changes lengths/areas (unless scale factor is 1), rotation preserves distances/angles, and mirroring preserves distances/angles but flips handedness (left/right).

⚠️ Important modelling constraint: in most CAD systems, you must specify the reference geometry for each command (origin/point, axis, plane). Different reference choices produce different results even with the same scale factor or angle.

Key geometric terms you’ll use repeatedly

• Reference point
• Axis
• Plane
• Uniform scaling
• Non-uniform scaling

Mermaid view of the three operations (2D illustration):

(a) Scaling — what it means and how to specify it

Scaling changes each point’s position relative to a reference location.

1) Uniform vs. non-uniform scaling

Let a model point be $p=(x,y,z)$ and the reference point be $c=(c_x,c_y,c_z)$.

Uniform scaling by factor $s$: $p' = c + s(p-c)$

This multiplies all displacement vectors from $c$ by the same factor $s$. Distances are scaled by $|s|$.

Non-uniform scaling with factors $(s_x,s_y,s_z)$: $p' = c + (s_x(x-c_x),\; s_y(y-c_y),\; s_z(z-c_z))$

This changes proportions unless all factors are equal.

Modelling implication: non-uniform scaling can turn squares into rectangles, circles into ellipses (in the scaled plane), and can also make normals/face appearance behave differently depending on how the CAD kernel updates geometry.

2) What the CAD “Scaling command” usually asks for

Most geometric modellers require:

• Scale factor
• Reference point
• Optional: Copy vs. move
• Optional: Scaling type

3) Invariance and what changes

• Collinearity is preserved: if points lie on a line, their scaled images still lie on a line.
• Ratios of distances along the same ray from the reference point remain consistent.
• Areas scale by $s^2$ (uniform) and volumes by $s^3$ (uniform)—useful for intuition in engineering and mass-properties workflows.

(b) Mirror — reflection across a line/plane

Mirror is a reflection transform. In 3D it reflects points across a plane; in 2D it reflects across a line.

1) How reflection works conceptually

Choose a mirror plane with unit normal vector $n$ and offset such that the plane contains points with equation: $n\cdot(x - p_0)=0$ for some point $p_0$ on the plane.

Then for any point $p$, the mirrored point $p'$ is: $p' = p - 2\big(n\cdot(p-p_0)\big)\,n$

Geometric meaning:

• Decompose $p-p_0$ into components parallel and perpendicular to the plane.
• Flip the perpendicular component’s sign; the parallel component stays.

2) What the CAD “Mirror command” typically asks for

• A mirror plane (or mirror line in 2D), defined by:
  • a face plane,
  • a construction plane,
  • a selected axis + plane choice,
  • or 3 points that determine a plane (system-dependent).
• Optionally: Copy vs. edit-in-place.
• Optional: whether to “mirror with respect to” a feature coordinate system or global coordinates.

3) What stays the same, and what flips

• Distances and angles are preserved (rigid reflection).
• Orientation is reversed: a surface normal may appear flipped, and handedness changes (e.g., left-hand vs. right-hand).
• If you mirror a component and later pattern/assemble it, confirm mate/constraint conventions and normal directions.

(c) Rotation — spinning about an axis by an angle

Rotation rotates the model around an axis by a specified angle $\theta$.

1) Rotation in 3D: axis-angle idea

If the rotation axis is defined by a unit direction vector $u$ and the rotation passes through a reference point $c$, then:

• Translate so the axis passes through the origin.
• Apply rotation.
• Translate back.

A common compact form uses Rodrigues’ rotation formula for rotating a vector $v$ about unit axis $u$: $v' = v\cos\theta + (u\times v)\sin\theta + u(u\cdot v)(1-\cos\theta)$

For a point $p$: $p' = c + R_u(\theta)(p-c)$

2) What CAD usually asks for

• Rotation axis
• Angle $\theta$ (and sometimes:
  • direction (clockwise vs. counterclockwise),
  • whether the axis is inferred from two points, an edge, or a coordinate system)
• Reference for the axis location (origin, selected edge, sketch line)
• Optional: copy vs. edit.

3) Rotation preserves metric properties

• Distances and angles remain unchanged (rotation is an isometry).
• A circle stays a circle (its plane orientation changes, but radius remains constant).
• If you rotate about a wrong axis or wrong point on the axis, the part will “orbit” instead of spinning in place.

Quick comparison: Scaling vs. Mirror vs. Rotation

Summary table

Mermaid conceptual sketch: