- #1
SW VandeCarr
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Can a triangle be smoothly transformed to a circle?
SW VandeCarr said:Can a triangle be smoothly transformed to a circle?
SW VandeCarr said:Either this question is too easy, too dumb or too hard. I'd just like an answer. I've found two two possibly relevant theorems:
Hironaka: Every analytic space Y admits a resolution of singularities: there is a smooth manifold X and a proper map f:X->Y such that f is an isomorphism except over singular points.
Castelnuovo, Enriques: Every singular surface has a unique minimal resolution of singularities.
What exactly is a "minimal resolution"? It suggests to me that the singular points of a triangle cannot be "eliminated" by a smooth transformation, but only arbitrarily "minimized". Is this correct?
Singular points are points on a graph or equation where the function is not defined or is undefined. They can also be points where the function is discontinuous or has an infinite value.
Singular points can be identified by finding points where the function is undefined, discontinuous, or has a vertical tangent line. They can also be identified by solving for points where the denominator of a fraction is equal to zero.
Transformations such as translation, dilation, and rotation can involve singular points. These transformations can affect the position, size, and orientation of the graph, potentially creating or removing singular points.
Singular points can cause the graph to have breaks or holes, or to have a vertical or horizontal asymptote. They can also cause the graph to be discontinuous or have sharp turns.
In mathematical models, singular points can be handled by using limits or piecewise functions to represent the discontinuity or undefined points. They can also be approximated by using numerical methods to calculate the value of the function at the singular point.