【PEP】Junior High School Math, Grade 9, Part 2: The Art of Light and Shadow

Imagine ancient Egypt thousands of years ago, where massive pyramids cast long shadows under the blazing sun. This was not just a measure of time—it marked the beginning of humanity's observation of spatial relationships. From the ancientsundialto modern engineering blueprints, humans gradually transformed the intuitive 'shadow' into rational 'mathematical language'.

1. Scientific Definition of Projection

Projection is not merely a simple 'black shadow,' but a geometric record formed when light passes through an object and lands on a plane. It consists of three core elements:

Projection Line: The incident rays of light.
Projection Plane: The plane where the shadow lies (e.g., ground, wall).
Projection: The resulting shape on the projection plane.

Evolution of Classification

Central Projection: A projection formed by rays emanating from a single point (point source, like a bulb or flame). Characterized by 'larger up close, smaller from afar,' commonly used in perspective drawing.

Parallel Projection: A projection formed by parallel rays (like sunlight from a distant source). It is further divided into:

Oblique Projection: The projection lines are slanted relative to the projection plane.
Orthographic Projection: The projection linesperpendicularto the projection plane. This forms the foundation for drawing engineering drawings (three views).

2. The Foundation of Descriptive Geometry

Descriptive Geometrywas founded by French mathematicianGaspard Monge He studied how to accurately represent and reconstruct three-dimensional spatial structures on a two-dimensional plane using orthographic projection. Indeed, without orthographic projection, there would be no engineering language for modern precision manufacturing.

🎯 Core Principle

Studying views cannot be separated from studying projections. The essence of orthographic projection lies in the perpendicular alignment between light rays and the projection plane, preserving a figure's 'true dimensions' without distortion.

QUESTION 1

Xiao Hua took several photos at Tiananmen Square at different times. Which photo was taken in the afternoon?

Shadow pointing directly west

Shadow pointing directly east

Shadow extremely short and directly beneath

Shadow pointing directly north

QUESTION 2

A projection formed by rays emanating from a single point (point source) is called:

Orthographic Projection

Oblique Projection

Central Projection

Parallel Projection

QUESTION 3

In parallel projection, if the projection lines are perpendicular to the projection plane, this type of projection is called:

Central Projection

Orthographic Projection

Perspective Projection

Oblique Projection

QUESTION 4

The mathematician known as the 'Father of Descriptive Geometry' who established the foundational theory of projection is:

Euclid

Gaspard Monge

Archimedes

Gauss

QUESTION 5

Which of the following phenomena is an example of central projection?

Shadows of trees under sunlight

A person's shadow under a streetlight

Shadow of a gnomon on a sundial

Shadow produced by parallel light from a spotlight

Deep Challenge: Light and Shadow Logic & Spatial Modeling

Q1. [Matching] Connect the following objects/phrases with their corresponding projection characteristics.

1. Animal shapes in shadow puppetry —— A. Circular projection
2. Cylinder under sunlight (vertical illumination of base) —— B. Rays from a point source (central projection)
3. Triangle ruler under a spotlight —— C. Rectangular projection (specific angle)
4. Sphere illuminated perpendicularly —— D. Shadow outline
5. Cylinder under sunlight (parallel illumination of side) —— E. Parallel projection

Reference Answers:
1-D (hand gesture forms shadow outline)
2-A (base projects as a circle)
3-B (spotlight/point source forms central projection)
4-A (sphere always projects as a circle)
5-C (side projects as a rectangle)

Q2. [Comprehensive Analysis] Figure 29.1-5 shows three projections of a triangle ruler. Answer:
(1) What is the difference between the projection lines in Figure (1) and Figures (2)(3)?
(2) How do the projection lines in Figures (2)(3) relate to the projection plane differently?

Model Solution:
(1) The projection lines in Figure (1) originate from a single point, belonging toCentral Projection; the projection lines in Figures (2)(3) are mutually parallel, belonging toParallel Projection.
(2) In Figure (2), the projection lines are slanted relative to the projection plane, making itOblique Projection; in Figure (3), the projection lines are perpendicular to the projection plane, making itOrthographic Projection.

Q3. [Spatial Imagination] Observe a regular pentagonal prism. When light shines vertically downward, what are the orthographic projections of each face?

Reference Answers:
1. Top and bottom bases: Since they are perpendicular to the light, their orthographic projections are congruent with the baseregular pentagon.
2. Five lateral faces: Since the lateral edges are perpendicular to the base, the faces are parallel to the light, so their orthographic projections all reduce toline segments.

Q4. [Engineering Practice] Given the three views of a cone and an L-shaped composite body, consider:
(1) What is the shape of the cone's net?
(2) If modeling with a potato, how would you cut it based on the three views?

Practical Guidance:
(1) The cone's net consists of onesector(lateral surface) and onecircle(base).
(2) When constructing the model, first cut the outer contour of the base according to the top view, then use the front and left views to determine height and slope for refining the sides, ensuring all three views align with the given blueprint.