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Nandakumar R
Nandakumar R
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Here we consider uniform planar convex regions and lines that lie on that same plane.

The following can be easily proved using perpendicular axes theorem and intermediate value theorem:

Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least one pair of mutually perpendicular lines that pass thru $P$ and lie on the same plane as $C$ such that the moment of inertia (MI) of $C$ about both lines are equal.

Symmetries in $C$ can lead to some points on $C$ being 'special' - such that w. r. to more than 2 lines thru them, MI of $C$ is the same; the center of a circular disk is obviously very special.

Question: Is it possible that for any planar convex $C$, there is at least one special point - a point with more than 2 lines passing through it giving the same MI for $C$? If such a special point always exists for any $C$, in the case when $C$ is a triangle, will it coincide with any known center of the triangle?

Remark: Above lemma won't hold for other moments - those defined with powers of distance from the axis different from quadratic. But analogous questions about concurrent lines yielding the same moment for $C$ could be asked - maybe with special points having only 2 lines of same moment passing thru them.

Note (January 2024): This question does overlap with this earlier question: On moments of inertia of planar and 3D convex bodies but the bit on the possible triangle center seems new.

Here we consider uniform planar convex regions and lines that lie on that same plane.

The following can be easily proved using perpendicular axes theorem and intermediate value theorem:

Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least one pair of mutually perpendicular lines that pass thru $P$ and lie on the same plane as $C$ such that the moment of inertia (MI) of $C$ about both lines are equal.

Symmetries in $C$ can lead to some points on $C$ being 'special' - such that w. r. to more than 2 lines thru them, MI of $C$ is the same; the center of a circular disk is obviously very special.

Question: Is it possible that for any planar convex $C$, there is at least one special point - a point with more than 2 lines passing through it giving the same MI for $C$?

Remark: Above lemma won't hold for other moments - those defined with powers of distance from the axis different from quadratic. But analogous questions about concurrent lines yielding the same moment for $C$ could be asked - maybe with special points having only 2 lines of same moment passing thru them.

The following can be easily proved using perpendicular axes theorem and intermediate value theorem:

Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least one pair of mutually perpendicular lines that pass thru $P$ and lie on the same plane as $C$ such that the moment of inertia (MI) of $C$ about both lines are equal.

Symmetries in $C$ can lead to some points on $C$ being 'special' - such that w. r. to more than 2 lines thru them, MI of $C$ is the same; the center of a circular disk is obviously very special.

Question: Is it possible that for any planar convex $C$, there is at least one special point - a point with more than 2 lines passing through it giving the same MI for $C$? If such a special point always exists for any $C$, in the case when $C$ is a triangle, will it coincide with any known center of the triangle?

Remark: Above lemma won't hold for other moments - those defined with powers of distance from the axis different from quadratic. But analogous questions about concurrent lines yielding the same moment for $C$ could be asked - maybe with special points having only 2 lines of same moment passing thru them.

Note (January 2024): This question does overlap with this earlier question: On moments of inertia of planar and 3D convex bodies but the bit on the possible triangle center seems new.

Source Link
Nandakumar R
Nandakumar R
  • 6k
  • 3
  • 7
  • 20

On 'special' points on uniform planar convex regions defined in terms of moment of inertia

Here we consider uniform planar convex regions and lines that lie on that same plane.

The following can be easily proved using perpendicular axes theorem and intermediate value theorem:

Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least one pair of mutually perpendicular lines that pass thru $P$ and lie on the same plane as $C$ such that the moment of inertia (MI) of $C$ about both lines are equal.

Symmetries in $C$ can lead to some points on $C$ being 'special' - such that w. r. to more than 2 lines thru them, MI of $C$ is the same; the center of a circular disk is obviously very special.

Question: Is it possible that for any planar convex $C$, there is at least one special point - a point with more than 2 lines passing through it giving the same MI for $C$?

Remark: Above lemma won't hold for other moments - those defined with powers of distance from the axis different from quadratic. But analogous questions about concurrent lines yielding the same moment for $C$ could be asked - maybe with special points having only 2 lines of same moment passing thru them.