2006-12-16, 08:35

i have found a way to parametrize a circle whose radius is unknown using only a part of the arc. it is helpful if you draw this as you read.

we place the arc so that it is concave directly down, then connect the two endpoints. we take a tangent line to the circle where it starts on the right endpoint. we call the angle it makes with the horizontal line joining the two endpoints: "alpha". we make a perpendicular bisector through the horizontal line, and measure the distance from the point of intersection of the two lines, to the intersection of the bisector and the curve. we call this distance "h"

now that we have defined h and alpha, we need to relate these two bits to the radius.

from geometry, we know that any tangent to a circle is perpendicular to the radius.

since we assumed the arc is part of a circle (a top portion), we can call the horizontal line "r sin(@)" where r is the unknown radius of the circle because it is that many units above the origin of the circle.

so now, h+r sin(@)=r --> h=r[1-sin(@)]

and since we measured (approximated) the height h of the arc, we can solve for r --> r=h/[1-sin(@)]

but we do not know the value for @.

so we now need a relationship between alpha and @.

let us introduce an x,y coordinate system, and have it centered at the origin of the circle. again, from geometry, since the x axis is parallel to r sin(@), and the radial line r, intersects these parallel lines, the angle between r sin(@) and the radial line will be the same as @.

now we will call this angle "gamma"

once more, from geometry, the right angle made from the radial line and the tangent is made of alpha and gamma. so these must add to pi/2

and since gamma=@, pi/2-alpha=@ --> alpha=pi/2-@

now we can plug in a value for alpha that is in direct correlation to @ which the radius depends on.

our formula for radius becomes: r=h/[1-sin(pi/2-@)]

and from trigonometry, sin(pi/2-**)=cos(**)

so this becomes: r=h/[1-cos(alpha)]

so we can now compute radius.

but we need a value for alpha. there are two ways that work as well as you want/need them to: ruler, or ruler and protractor. you can use a protractor to measure the angle, or you can draw a triangle with the ruler and use a ratio of>> 1/2 the horizontal line:hypotenuse.

plug in your knowns and let it rip.

once you are convinced you have a pretty good approximation of the radius, you can parametrize a circle and find anything you want about it.

r(t)=< cos(t) h/[1-cos(alpha)], sin(t) h/[1-cos(alpha)] >

or r(t)= h/[1-cos(alpha)] < cos(t), sin(t) >

and the curvature will be: 1/radius of circle --> [1-cos(alpha)]/h

there you have it. you can also use a part of the arc of a curve to approximate its curvature at a point using the same method.

play around with it and let me know if yo uguys find anything interesting.

-joe

we place the arc so that it is concave directly down, then connect the two endpoints. we take a tangent line to the circle where it starts on the right endpoint. we call the angle it makes with the horizontal line joining the two endpoints: "alpha". we make a perpendicular bisector through the horizontal line, and measure the distance from the point of intersection of the two lines, to the intersection of the bisector and the curve. we call this distance "h"

now that we have defined h and alpha, we need to relate these two bits to the radius.

from geometry, we know that any tangent to a circle is perpendicular to the radius.

since we assumed the arc is part of a circle (a top portion), we can call the horizontal line "r sin(@)" where r is the unknown radius of the circle because it is that many units above the origin of the circle.

so now, h+r sin(@)=r --> h=r[1-sin(@)]

and since we measured (approximated) the height h of the arc, we can solve for r --> r=h/[1-sin(@)]

but we do not know the value for @.

so we now need a relationship between alpha and @.

let us introduce an x,y coordinate system, and have it centered at the origin of the circle. again, from geometry, since the x axis is parallel to r sin(@), and the radial line r, intersects these parallel lines, the angle between r sin(@) and the radial line will be the same as @.

now we will call this angle "gamma"

once more, from geometry, the right angle made from the radial line and the tangent is made of alpha and gamma. so these must add to pi/2

and since gamma=@, pi/2-alpha=@ --> alpha=pi/2-@

now we can plug in a value for alpha that is in direct correlation to @ which the radius depends on.

our formula for radius becomes: r=h/[1-sin(pi/2-@)]

and from trigonometry, sin(pi/2-**)=cos(**)

so this becomes: r=h/[1-cos(alpha)]

so we can now compute radius.

but we need a value for alpha. there are two ways that work as well as you want/need them to: ruler, or ruler and protractor. you can use a protractor to measure the angle, or you can draw a triangle with the ruler and use a ratio of>> 1/2 the horizontal line:hypotenuse.

plug in your knowns and let it rip.

once you are convinced you have a pretty good approximation of the radius, you can parametrize a circle and find anything you want about it.

r(t)=< cos(t) h/[1-cos(alpha)], sin(t) h/[1-cos(alpha)] >

or r(t)= h/[1-cos(alpha)] < cos(t), sin(t) >

and the curvature will be: 1/radius of circle --> [1-cos(alpha)]/h

there you have it. you can also use a part of the arc of a curve to approximate its curvature at a point using the same method.

play around with it and let me know if yo uguys find anything interesting.

-joe