12.2 Three-Dimensional Views and Cross-Sections

Section 12.2: Three-Dimensional Views and Cross-Sections

Spatial representation and planar cuts of 3D figures are core testing points in Gaokao Mathematics. Mastering the relationship between three-dimensional solids and their two-dimensional orthographic projections or planar cross-sections requires systematic geometric rules and precise visualization techniques.

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1. Principles of Orthographic Projections (三视图)

1.1 The Three Projection Planes and View Alignment

An orthographic projection projects a 3D figure orthogonally onto three mutually perpendicular planes:

1. Front View (正视图/主视图): Projected onto the vertical plane $V$. Reflects the solid's width (length) and height.
2. Side View (侧视图/左视图): Projected onto the profile plane $W$ from the left. Reflects the solid's depth (width) and height.
3. Top View (俯视图): Projected onto the horizontal plane $H$ from above. Reflects the solid's width (length) and depth (width).

1.2 Core Rules of Projection Alignment ("九字真言")

• 长对正 (Length Aligned): The length in the Front View and Top View must align vertically.
• 高平齐 (Height Level): The height in the Front View and Side View must match horizontally.
• 宽相等 (Width Equal): The width (depth) in the Top View and Side View must be identical.

1.3 Solid vs. Dashed Line Representation

• Solid Line (实线): Represents boundary edges or intersections that are directly visible from the line of sight.
• Dashed Line (虚线): Represents hidden boundary edges or internal structural lines blocked by opaque front features.

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2. Reconstructing 3D Solids from Three Views (三视图还原几何体)

To compute volumes, surface areas, or edge lengths from orthographic views, use the Bounding Cuboid Grid Method (长方体网格法):

1. Step 1: Construct the Bounding Box: Draw a standard rectangular cuboid $ABCD-A_1B_1C_1D_1$ whose dimensions $(L, W, H)$ match the maximum dimensions given in the three views.
2. Step 2: Plot View Features on Faces:
   • Overlay the Front View onto the front face $ABB_1A_1$.
   • Overlay the Side View onto the left face $ADD_1A_1$.
   • Overlay the Top View onto the top face $A_1B_1C_1D_1$ or base $ABCD$.
3. Step 3: Connect Vertices and Prune:
   • Identify candidate 3D vertices by finding intersection points of projecting lines.
   • Eliminate vertices and edges that produce extra projection lines not present in the given views.
   • Verify solid vs. dashed lines for all three projections.

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3. Planar Cross-Sections of 3D Solids (截面问题)

3.1 Definition and Core Principles

A cross-section (截面) is the planar polygon formed by the intersection of a cutting plane $\alpha$ with the faces of a polyhedron.

• Intersection Theorem for Parallel Planes: If a cutting plane $\alpha$ intersects two parallel planes $\beta_1$ and $\beta_2$ along lines $l_1$ and $l_2$, then $l_1 \parallel l_2$.
• Maximum Polygon Sides: A cutting plane intersecting a polyhedron with $F$ faces can yield a cross-sectional polygon with at most $F$ sides. For a cube ($F = 6$), cross-sections can be triangles, quadrilaterals, pentagons, or hexagons, but never heptagons ($7$) or octagons ($8$).

3.2 Sphere and Cone Cross-Sections

1. Cross-Section of a Sphere: Any planar cut of a sphere with radius $R$ at distance $d$ ($0 \le d < R$) from the center $O$ is a circle with radius: $$r_{\text{section}} = \sqrt{R^2 - d^2}$$ $$\text{Area of Section: } S_{\text{section}} = \pi (R^2 - d^2)$$
2. Cross-Sections of a Cone:
   • Plane $\perp$ axis $\implies$ Circle.
   • Plane passing through apex $\implies$ Isosceles Triangle.
   • Plane parallel to one generator $\implies$ Parabola segment.
   • Plane inclined to axis (not passing through apex) $\implies$ Ellipse or Hyperbola branch.

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4. Worked Gaokao Exam Examples

Example 1: Volume and Edge Analysis from Three Views

Problem: The three views of a geometric solid built inside a unit cube of edge length $1$ are shown:

• Front View: A right isosceles triangle with legs $1, 1$.
• Side View: A right isosceles triangle with legs $1, 1$.
• Top View: A square of side length $1$.

Determine the shape of the solid, compute its volume $V$, and find the maximum edge length among all its edges.

Detailed Solution:

1. Reconstruct the 3D Solid: Place the solid inside a unit cube $ABCD-A_1B_1C_1D_1$ with base $ABCD$ on the bottom.
   • Top View is a square of side $1 \implies$ The projection on the base $ABCD$ covers the entire square $ABCD$.
   • Front View and Side View are right triangles with legs $1 \implies$ All lateral faces slope to a single apex $P$ located directly above vertex $A$, i.e., apex $P = A_1$.
   • The solid is a square pyramid $A_1-ABCD$, where lateral edge $A_1A \perp$ base $ABCD$.
2. Compute Volume: $$V = \frac{1}{3} S_{\text{base}} \cdot h = \frac{1}{3} \cdot (1 \times 1) \cdot 1 = \frac{1}{3}$$
3. Analyze Edge Lengths:
   • Base edges: $AB = BC = CD = DA = 1$.
   • Vertical edge: $A_1A = 1$.
   • Adjacent lateral edges: $A_1B = \sqrt{A_1A^2 + AB^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$, and $A_1D = \sqrt{1^2 + 1^2} = \sqrt{2}$.
   • Longest lateral diagonal edge: $A_1C = \sqrt{A_1A^2 + AC^2} = \sqrt{1^2 + (\sqrt{2})^2} = \sqrt{3}$.
   • Maximum edge length is $\sqrt{3}$.

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Example 2: Planar Cross-Section of a Sphere

Problem: A sphere $O$ has radius $R = 5$. A cutting plane $\alpha$ passes through a point $P$ inside the sphere such that the distance from the sphere center $O$ to point $P$ is $OP = 3$. Find the minimum possible area and maximum possible area of the circular cross-section formed by $\alpha$ passing through $P$.

Detailed Solution:

1. Maximum Section Area: Occurs when plane $\alpha$ passes through the sphere center $O$ (great circle, $d = 0$). Since $P$ lies on $\alpha$, a plane passing through $O$ and $P$ has distance $d_{\min} = 0$. $$S_{\max} = \pi R^2 = \pi (5^2) = 25\pi$$
2. Minimum Section Area: Occurs when plane $\alpha$ is perpendicular to line segment $OP$. In this case, the distance from center $O$ to plane $\alpha$ is $d_{\max} = OP = 3$. $$r_{\min} = \sqrt{R^2 - d_{\max}^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$$ $$S_{\min} = \pi (r_{\min})^2 = \pi (4^2) = 16\pi$$
3. The cross-sectional area $S$ satisfies $16\pi \le S \le 25\pi$.

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5. Reconstruction Workflow Visualization