Revisions to Drawing a smooth curve with the least maximum curvature from a point within a circle to its boundary

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edited Oct 11 at 18:58

2080

201
3
14

Given

a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

a start point, at radial distance $p$ (with $ 0 < p < r$) from the origin, and angle $\alpha$
an end point on the circle boundary at radial distance $r$ from the origin and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

it starts at the start point, aligned with angle $\alpha$ (radially outwards)
it ends at the end point, aligned with angle $\beta$ (radially outwards)
the curve stays within the circle area (intersects the circle boundary only once, at the end point)
between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
the maximum curvature is minimized

?

Without loss of generality, the angle $\alpha$ can be assumed to be 0, because only the angle difference between $\alpha$ and $\beta$ matters.

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there only works when $ p = 0 $.

An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).

Given

a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

a start point, at radial distance $p$ (with $ 0 < p < r$) from the origin, and angle $\alpha$
an end point on the circle boundary at radial distance $r$ from the origin and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

it starts at the start point, aligned with angle $\alpha$ (radially outwards)
it ends at the end point, aligned with angle $\beta$ (radially outwards)
the curve stays within the circle area (intersects the circle boundary only once, at the end point)
between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
the maximum curvature is minimized

?

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there only works when $ p = 0 $.

An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).

Given

a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

a start point, at radial distance $p$ (with $ 0 < p < r$) from the origin, and angle $\alpha$
an end point on the circle boundary at radial distance $r$ from the origin and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

it starts at the start point, aligned with angle $\alpha$ (radially outwards)
it ends at the end point, aligned with angle $\beta$ (radially outwards)
the curve stays within the circle area (intersects the circle boundary only once, at the end point)
between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
the maximum curvature is minimized

?

Without loss of generality, the angle $\alpha$ can be assumed to be 0, because only the angle difference between $\alpha$ and $\beta$ matters.

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there only works when $ p = 0 $.

An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).

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edited Oct 11 at 18:23

2080

201
3
14

Given

a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

a start point, at radial distance $p$ (with $ 0 < p < r$) from the origin, and angle $\alpha$
an end point on the circle boundary at radial distance $r$ from the origin and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

it starts at the start point, aligned with angle alpha$\alpha$ (radially outwards)
it ends at the end point, aligned with angle beta$\beta$ (radially outwards)
the curve stays within the circle area (intersects the circle boundary only once, at the end point)
between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
the maximum curvature is minimized

?

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there only works when $ p = 0 $.

An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).

Given

a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

a start point, at radial distance $p$ (with $ 0 < p < r$) from the origin, and angle $\alpha$
an end point on the circle boundary at radial distance $r$ from the origin and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

it starts at the start point, aligned with angle alpha (radially outwards)
it ends at the end point, aligned with angle beta (radially outwards)
the curve stays within the circle area (intersects the circle boundary only once, at the end point)
between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
the maximum curvature is minimized

?

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there only works when $ p = 0 $.

An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).

Given

a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

a start point, at radial distance $p$ (with $ 0 < p < r$) from the origin, and angle $\alpha$
an end point on the circle boundary at radial distance $r$ from the origin and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

it starts at the start point, aligned with angle $\alpha$ (radially outwards)
it ends at the end point, aligned with angle $\beta$ (radially outwards)
the curve stays within the circle area (intersects the circle boundary only once, at the end point)
between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
the maximum curvature is minimized

?

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there only works when $ p = 0 $.

An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).

added 16 characters in body

Source Link

edited Oct 11 at 14:47

2080

201
3
14

Given

a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

a start point, at radial distance $p$ ($< r$with $ 0 < p < r$) for the radial distance of the point from the origin, and angle $\alpha$
an end point on the circle boundary atat radial distance $r$ from the origin and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

it starts at the start point, aligned with angle alpha (radially outwards)
it ends at the end point, aligned with angle beta (radially outwards)
the curve stays within the circle area (intersects the circle boundary only once, at the end point)
between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
the maximum curvature is minimized

?

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there does not work hereonly works when $ p = 0 $.

An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).

Given

a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

a start point, $p$ ($< r$) for the radial distance of the point from the origin, and angle $\alpha$
an end point on the circle boundary at radial distance $r$ and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

it starts at the start point, aligned with angle alpha (radially outwards)
it ends at the end point, aligned with angle beta (radially outwards)
the curve stays within the circle area (intersects the circle boundary only once, at the end point)
between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
the maximum curvature is minimized

?

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there does not work here.

An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).

Given

a circle with radius $r$ centered on the origin at (0,0)

and, in polar coordinates:

a start point, at radial distance $p$ (with $ 0 < p < r$) from the origin, and angle $\alpha$
an end point on the circle boundary at radial distance $r$ from the origin and angle $\beta$

how can I construct a curve segment that satisfies the following constraints:

it starts at the start point, aligned with angle alpha (radially outwards)
it ends at the end point, aligned with angle beta (radially outwards)
the curve stays within the circle area (intersects the circle boundary only once, at the end point)
between the start and end points, the radial distance of points on the curve from the (0,0) origin is non-decreasing (ideally, strictly increasing)
the maximum curvature is minimized

?

Does a function or algorithm for this exist?

This is a variation of this question that only considered the start point being placed at the center of the circle. The solution there only works when $ p = 0 $.

An example of curves that fulfill the first three constraints, and most the fourth, but only the straight lines the last (and happen to have their start point at the center).