Module IV·Article I·~1 min read
Ruled Surfaces and Their Properties
Quadric Surfaces
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Ruled Surfaces
Ruled Surfaces
A ruled surface is one through each point of which passes a straight line lying entirely on the surface. These lines are called generators.
Cylinders and cones are obviously ruled. More surprisingly, the one-sheeted hyperboloid and the hyperbolic paraboloid are also ruled.
One-sheeted Hyperboloid
x²/a² + y²/b² − z²/c² = 1.
Two families of generators: through every point pass two lines from different families. Any two generators from one family are skew.
Constructive significance: hyperbolic towers are built from straight bars (material savings + rigidity). The Shukhov Tower in Moscow (1922) was the first example.
Hyperbolic Paraboloid (Saddle)
z/c = x²/a² − y²/b² (often given as z = xy).
Two families of straight line generators. The surface is “convex” in one direction and “concave” in the other.
Application in architecture: the hyperbolic paraboloid can be built from straight beams. "Thin shells" in 20th century architecture (Felix Candela).
Surfaces of Revolution
When rotating a curve f(y, z) = 0 around the z-axis: replace y with √(x²+y²).
Torus: rotation of the circle (y−R)² + z² = r² about the z-axis: (√(x²+y²)−R)² + z² = r².
Torus in topology: the product of two circles S¹ × S¹. Fundamental group: ℤ × ℤ.
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