[math-fun] Circle question
Consider a circle whose diameter lies on a fixed axis line passing obliquely through a plane (not perpendicular to the plane) Now rotate the circle about the axis line subject to the following conditions: - The plane position is fixed. - The circle radius is fixed. - The axis is parallel to its initial position. - The circle is tangent to the plane, on the same side of the plane as its initial position. Can the circle be made to move without slippage along the plane? What is the locus of the point of circle-plane contact?
I must be missing something. Let P be the original plane, and Q be the plane containing the axis that is perpendicular to P. Can't you just move the axis in the direction perpendicular to Q, rolling the circle as you go, so that the circle-plane contact describes a straight line? More generally, move the axis any way you like, keep the circle in a plane parallel to its original position. Have the circle rotate so that the speed of rotation matches the speed at which the circle/plane intersection moves. Isn't this "rolling without slipping"? Andy On Sun, Sep 9, 2012 at 12:01 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a circle whose diameter lies on a fixed axis line passing obliquely through a plane (not perpendicular to the plane)
Now rotate the circle about the axis line subject to the following conditions:
- The plane position is fixed. - The circle radius is fixed. - The axis is parallel to its initial position. - The circle is tangent to the plane, on the same side of the plane as its initial position.
Can the circle be made to move without slippage along the plane? What is the locus of the point of circle-plane contact?
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Andy, I'm not sure I understand the question, either, but I think the axis is supposed to be *not* perpendicular to P. Rather it's supposed to be "oblique" to P. --Dan On 2012-09-09, at 5:25 PM, Andy Latto wrote:
I must be missing something. Let P be the original plane, and Q be the plane containing the axis that is perpendicular to P. Can't you just move the axis in the direction perpendicular to Q, rolling the circle as you go, so that the circle-plane contact describes a straight line?
More generally, move the axis any way you like, keep the circle in a plane parallel to its original position. Have the circle rotate so that the speed of rotation matches the speed at which the circle/plane intersection moves. Isn't this "rolling without slipping"?
Andy
On Sun, Sep 9, 2012 at 12:01 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a circle whose diameter lies on a fixed axis line passing obliquely through a plane (not perpendicular to the plane)
Now rotate the circle about the axis line subject to the following conditions:
- The plane position is fixed. - The circle radius is fixed. - The axis is parallel to its initial position. - The circle is tangent to the plane, on the same side of the plane as its initial position.
Can the circle be made to move without slippage along the plane? What is the locus of the point of circle-plane contact?
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On Sun, Sep 9, 2012 at 10:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Andy,
I'm not sure I understand the question, either, but I think the axis is supposed to be *not* perpendicular to P. Rather it's supposed to be "oblique" to P.
Yes, I know. But there is a direction in P perpendicular to the ray, and that's the direction I propose to move. There is still a plane Q containing the ray and perpendicular to P. Andy
--Dan
On 2012-09-09, at 5:25 PM, Andy Latto wrote:
I must be missing something. Let P be the original plane, and Q be the plane containing the axis that is perpendicular to P. Can't you just move the axis in the direction perpendicular to Q, rolling the circle as you go, so that the circle-plane contact describes a straight line?
More generally, move the axis any way you like, keep the circle in a plane parallel to its original position. Have the circle rotate so that the speed of rotation matches the speed at which the circle/plane intersection moves. Isn't this "rolling without slipping"?
Andy
On Sun, Sep 9, 2012 at 12:01 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a circle whose diameter lies on a fixed axis line passing obliquely through a plane (not perpendicular to the plane)
Now rotate the circle about the axis line subject to the following conditions:
- The plane position is fixed. - The circle radius is fixed. - The axis is parallel to its initial position. - The circle is tangent to the plane, on the same side of the plane as its initial position.
Can the circle be made to move without slippage along the plane? What is the locus of the point of circle-plane contact?
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-- Andy.Latto@pobox.com
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-- Andy.Latto@pobox.com
The circle is fixed to the axis along a diameter. It does not rotate in its own plane. It is the axis that is rotated, the circle and its plane spin around this axis. The circle is kept in contact with the plane, and "walks" along the plane. The circle is constrained to touch the plane at a point (contact point), the axis constrained to remain parallel to its initial orientation. If the axis is perpendicular to the plane, the circle spins on the contact point like a coin spun on a tabletop, and the locus of the contact point is just the contact point itself. If the axis is oblique to the plane, the point of contact should describe a periodic curve along the plane, I am thinking with cusps, similar to a cycloid. I am not sure if slippage is necessary, if so it would only be allowed in a direction tangent to the circle. On 9/10/2012 12:55 AM, Andy Latto wrote:
On Sun, Sep 9, 2012 at 10:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Andy,
I'm not sure I understand the question, either, but I think the axis is supposed to be *not* perpendicular to P. Rather it's supposed to be "oblique" to P. Yes, I know. But there is a direction in P perpendicular to the ray, and that's the direction I propose to move. There is still a plane Q containing the ray and perpendicular to P.
Andy
--Dan
On 2012-09-09, at 5:25 PM, Andy Latto wrote:
I must be missing something. Let P be the original plane, and Q be the plane containing the axis that is perpendicular to P. Can't you just move the axis in the direction perpendicular to Q, rolling the circle as you go, so that the circle-plane contact describes a straight line?
More generally, move the axis any way you like, keep the circle in a plane parallel to its original position. Have the circle rotate so that the speed of rotation matches the speed at which the circle/plane intersection moves. Isn't this "rolling without slipping"?
Andy
On Sun, Sep 9, 2012 at 12:01 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a circle whose diameter lies on a fixed axis line passing obliquely through a plane (not perpendicular to the plane)
Now rotate the circle about the axis line subject to the following conditions:
- The plane position is fixed. - The circle radius is fixed. - The axis is parallel to its initial position. - The circle is tangent to the plane, on the same side of the plane as its initial position.
Can the circle be made to move without slippage along the plane? What is the locus of the point of circle-plane contact?
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-- Andy.Latto@pobox.com
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So are we holding a lollipop at an angle against the plane, and twiddling the stick without changing the direction of the stick? On Mon, Sep 10, 2012 at 8:18 AM, David Wilson <davidwwilson@comcast.net>wrote:
The circle is fixed to the axis along a diameter. It does not rotate in its own plane. It is the axis that is rotated, the circle and its plane spin around this axis. The circle is kept in contact with the plane, and "walks" along the plane.
The circle is constrained to touch the plane at a point (contact point), the axis constrained to remain parallel to its initial orientation.
If the axis is perpendicular to the plane, the circle spins on the contact point like a coin spun on a tabletop, and the locus of the contact point is just the contact point itself.
If the axis is oblique to the plane, the point of contact should describe a periodic curve along the plane, I am thinking with cusps, similar to a cycloid.
I am not sure if slippage is necessary, if so it would only be allowed in a direction tangent to the circle.
On 9/10/2012 12:55 AM, Andy Latto wrote:
On Sun, Sep 9, 2012 at 10:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Andy,
I'm not sure I understand the question, either, but I think the axis is supposed to be *not* perpendicular to P. Rather it's supposed to be "oblique" to P.
Yes, I know. But there is a direction in P perpendicular to the ray, and that's the direction I propose to move. There is still a plane Q containing the ray and perpendicular to P.
Andy
--Dan
On 2012-09-09, at 5:25 PM, Andy Latto wrote:
I must be missing something. Let P be the original plane, and Q be the
plane containing the axis that is perpendicular to P. Can't you just move the axis in the direction perpendicular to Q, rolling the circle as you go, so that the circle-plane contact describes a straight line?
More generally, move the axis any way you like, keep the circle in a plane parallel to its original position. Have the circle rotate so that the speed of rotation matches the speed at which the circle/plane intersection moves. Isn't this "rolling without slipping"?
Andy
On Sun, Sep 9, 2012 at 12:01 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a circle whose diameter lies on a fixed axis line passing obliquely through a plane (not perpendicular to the plane)
Now rotate the circle about the axis line subject to the following conditions:
- The plane position is fixed. - The circle radius is fixed. - The axis is parallel to its initial position. - The circle is tangent to the plane, on the same side of the plane as its initial position.
Can the circle be made to move without slippage along the plane? What is the locus of the point of circle-plane contact?
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OK, I think I have an understanding of what you're asking from a physical perspective, but I'm having trouble coming up with a precise mathematical meaning of the "rolls without slipping" requirement. Part of this requirement is that the length of the path as measured as a curve in the plane is equal to the length as measured in arc length of the circle as the point of tangency varies (where the points on the axis are regarded as the "same" point on the circle as the circle is moved). But this doesn't restrict the path at all, other than its length. What's the other restriction conveyed by "rolls without slipping"? I'm not saying there isn't one, just that I can't figure out how to state it. Andy On Mon, Sep 10, 2012 at 8:18 AM, David Wilson <davidwwilson@comcast.net> wrote:
The circle is fixed to the axis along a diameter. It does not rotate in its own plane. It is the axis that is rotated, the circle and its plane spin around this axis. The circle is kept in contact with the plane, and "walks" along the plane.
The circle is constrained to touch the plane at a point (contact point), the axis constrained to remain parallel to its initial orientation.
If the axis is perpendicular to the plane, the circle spins on the contact point like a coin spun on a tabletop, and the locus of the contact point is just the contact point itself.
If the axis is oblique to the plane, the point of contact should describe a periodic curve along the plane, I am thinking with cusps, similar to a cycloid.
I am not sure if slippage is necessary, if so it would only be allowed in a direction tangent to the circle.
On 9/10/2012 12:55 AM, Andy Latto wrote:
On Sun, Sep 9, 2012 at 10:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Andy,
I'm not sure I understand the question, either, but I think the axis is supposed to be *not* perpendicular to P. Rather it's supposed to be "oblique" to P.
Yes, I know. But there is a direction in P perpendicular to the ray, and that's the direction I propose to move. There is still a plane Q containing the ray and perpendicular to P.
Andy
--Dan
On 2012-09-09, at 5:25 PM, Andy Latto wrote:
I must be missing something. Let P be the original plane, and Q be the plane containing the axis that is perpendicular to P. Can't you just move the axis in the direction perpendicular to Q, rolling the circle as you go, so that the circle-plane contact describes a straight line?
More generally, move the axis any way you like, keep the circle in a plane parallel to its original position. Have the circle rotate so that the speed of rotation matches the speed at which the circle/plane intersection moves. Isn't this "rolling without slipping"?
Andy
On Sun, Sep 9, 2012 at 12:01 PM, David Wilson <davidwwilson@comcast.net> wrote:
Consider a circle whose diameter lies on a fixed axis line passing obliquely through a plane (not perpendicular to the plane)
Now rotate the circle about the axis line subject to the following conditions:
- The plane position is fixed. - The circle radius is fixed. - The axis is parallel to its initial position. - The circle is tangent to the plane, on the same side of the plane as its initial position.
Can the circle be made to move without slippage along the plane? What is the locus of the point of circle-plane contact?
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-- Andy.Latto@pobox.com
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participants (4)
-
Allan Wechsler -
Andy Latto -
Dan Asimov -
David Wilson