A cylinder is swept out by straight vertical lines. A helicoid is also swept out by straight lines, but twists as the line rotates and rises. The cylinder has K=0; the helicoid has negative K. Straight generators do not guarantee intrinsic flatness.
A ruled patch has form X(s,t)=a(s)+t b(s), with b(s) nonzero. The parameter t moves along a generator while s selects the generator. Regularity requires (a′+tb′)×b≠0.
Predict
Predict which second derivative is simplest. Because X is affine in t, X_tt=0. That forces one coefficient of the second form to vanish and constrains the sign of K.
Predict whether the tangent developable of a space curve is regular along its original curve. Its generator is the tangent there, and the two coordinate derivatives coincide on that edge, creating a rank failure.
Experiment
Compare ruled examples.
Surface
Parametrization
Gaussian curvature
Plane
(s,t,0)
0
Cylinder
(cos s,sin s,t)
0
Helicoid
(t cos s,t sin s,s)
−1/(1+t²)²
For the helicoid, compute the cross product and its squared magnitude 1+t². Then compute the mixed second coefficient to verify the displayed K. The result is exact on the whole regular patch, not merely near the axis.
Understand
Since X_tt=0, the coefficient g=N·X_tt of II vanishes. Therefore
K=(eg−f²)/(EG−F²)=−f²/(EG−F²)≤0.
The denominator is positive by regularity. This proves every regular ruled patch has nonpositive Gaussian curvature.
Its mixed coefficient is f=N·b′. Substituting the normalized cross product gives
f=det(a′,b,b′)/||(a′+tb′)×b||.
The term involving t vanishes because b′ appears twice in its determinant. Thus K=0 exactly when det(a′,b,b′)=0 at the regular point. This is the developability criterion in the ruled setting.
For the tangent developable X(s,t)=γ(s)+tγ′(s), with γ unit speed and positive curvature, X_s=γ′+tγ″ and X_t=γ′. Their cross product is tγ″×γ′. It is nonzero for t≠0 and zero on t=0. The determinant criterion vanishes because a′=b, so the regular regions are developable while the original curve is a singular edge of this parametrization.
Apply
Repair “K=0 everywhere makes the surface globally a plane.” A cylinder is locally flat but has different global topology and closed circumferential geodesics. Even a local flatness conclusion requires the intrinsic curvature theorem or a direct unfolding construction, not just the word developable.
The determinant test is local and assumes a regular ruled patch. At a singular point the denominator vanishes and the curvature formula is not defined; assigning zero there by cancellation would erase the geometric issue.
For optional depth, cones and tangent developables can be unfolded locally away from their singular sets. Their global seams, vertices and self-intersections need separate treatment. The current proof establishes the curvature criterion, not a classification of every possible singular developable surface.
Continue
State the regular domain before plotting or computing a ruled surface’s curvature. Continue to local isometries and intrinsic metrics to formulate precisely what it means to bend without stretching.
Apply · explain · revise
Practice and reasoning
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For the helicoid X(u,v)=(u cos v,u sin v,v), find K at u=0.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Every ruled regular surface has
For X(s,t)=a(s)+t b(s), b(s)≠0, state and justify regularity at (s,t).
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Show a ruled surface has K=−f²/(EG−F²)≤0, where f=N·X_st.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show K=0 exactly when det(a′,b,b′)=0 at a regular ruled point.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For X(s,t)=γ(s)+tγ′(s) with a unit-speed curve of positive curvature, show regularity for t≠0 and explain the singular edge t=0.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
