Projection

Geometry
definition

Also known as: shadow, projected image, dimensional reduction

Grade 9-12

View on concept map

The image formed when points of a shape are mapped onto a lower-dimensional surface along parallel or converging rays. Basis for maps, blueprints, and visualizing higher dimensions.

Definition

The image formed when points of a shape are mapped onto a lower-dimensional surface along parallel or converging rays.

💡 Intuition

A shadow cast on the ground is a projection—a 3D object mapped down to a 2D silhouette.

🎯 Core Idea

Projection loses a dimension while preserving some structural information.

Example

A sphere's shadow is a circle; a cube's shadow can be a square or hexagon.

Notation

\text{proj}_{\ell}(P) denotes the projection of point P onto line \ell

🌟 Why It Matters

Basis for maps, blueprints, and visualizing higher dimensions.

💭 Hint When Stuck

Try shining a flashlight on an object from different angles and trace the shadow each time to see how projections change.

Formal View

Orthogonal projection of \vec{v} onto line with direction \hat{u}: \text{proj}_{\hat{u}}(\vec{v}) = (\vec{v} \cdot \hat{u})\hat{u}; onto subspace W: \text{proj}_W(\vec{v}) = \sum_{i} (\vec{v} \cdot \hat{e}_i)\hat{e}_i for orthonormal basis \{\hat{e}_i\} of W

Related Concepts

Dimension

🚧 Common Stuck Point

Different projections of the same 3D object can look very different depending on the angle of viewing.

⚠️ Common Mistakes

  • Thinking a projection preserves distances — projections typically distort lengths and areas
  • Confusing shadow projection (from a point source) with orthographic projection (parallel rays)
  • Assuming the projection uniquely determines the original object — many different 3D objects can cast the same 2D shadow

Frequently Asked Questions

What is Projection in Math?

The image formed when points of a shape are mapped onto a lower-dimensional surface along parallel or converging rays.

Why is Projection important?

Basis for maps, blueprints, and visualizing higher dimensions.

What do students usually get wrong about Projection?

Different projections of the same 3D object can look very different depending on the angle of viewing.

What should I learn before Projection?

Before studying Projection, you should understand: dimension.

Prerequisites

dimension

How Projection Connects to Other Ideas

To understand projection, you should first be comfortable with dimension.

← Back to Explorer