Equation to project any arbitrary point on the circumference of a circle to a line tangential to the circle.

[Akanimo]

Hi,

Assuming this is a perfect circle and a perfect line tangential to the circle, what would be the mathematical equation that projects any arbitrary point on the circumference of the circle to the tangential line? The desired expression has to have θ as a parameter but must not have sin(θ), cos(θ), tan(θ) or any of their reciprocals.

Thank you in advance.

 

[KlausST]

Hi,

for projeting a point .. I´d expected a coordinate system ... where we can see the origin (0/0).

what I seein your picture is the HORIZONTAL projection of a point to a VERTICAL line.
(and as long as we don´t have a coordinate system ... it does not matter whether the vertical ine is a tangente to the circle ... or far away from the circle)

But anyways:
* The horizontal value does not change (since the line is perfectly vertical), thus it is a constant value. Let´s call it X_P.
* the horizontal .. obviously is [ sine(phi) times R]. Let´s call it Y_P = R * sin(phi)

Now you say "sine()" is not allowed.
So this reduces your question to:

--> How to replace [ sin(phi) ] .. without using trigonometric functions.

Phi is part of a polar system .. X and Y are parts of a carthesic system.

And the only way to transform from polar to carthesic (I can remember) is using trigonometric functions.

Thus my question: WHY is "sin(phi)" not allowed?
--> I guess we need some context.

Klaus

 

[FvM]

Coordinate system not explicitely specified, also unclear if the problem is restricted to vertical tangent line. If not, you should use a less misleading example drawing.

Ar first sight, if you specify point cordinates by angle, you won't come away without using trigonometrical functions. But we should know expected point coordinate specification format.

 

[BradtheRad]

on the circumference of the circle

Measuring on the circumference itself is done in radians, yet this too is tied to the sine when projected to the tangential line.

Conversion to radians is θπ/180. The error is small within a few degrees of the perpendicular.

 

[BradtheRad]

There's the useful Taylor's series which can be adapted in approximating various mathematical functions. Mentioned in this article:

johndcook.com/blog/2010/07/27/sine-approximation-for-small-x/

 

[FvM]

Still waiting for a complete problem specification

 

[D.A.(Tony)Stewart]

Smells like a homework question.​

For the x coordinate of the tangent x_T using Center h,k with radius, r


x_T=h+r{x_T−h} / [SQRT]{(x_T−h)²+(y_T−k²)}

Similar for y_T

 

[Akanimo]

I apologize if my presentation was misleading. The problem is that an object will always be moving along the circular path - the circumference of the circle - at a constant velocity (although the velocity may change) which is obviously in the x-y plane. There is need to visualize and mimic this movement along the y-axis with respect to the angle it makes to the reference line without any consideration of x. The reference line is the radius perpendicular to the tangential line.

 

[D.A.(Tony)Stewart]

y(t) = sin (theta*t)

 

[BradtheRad]

visualize and mimic this movement along the y-axis with respect to the angle

Believe it or not, you're onto the classic definition of sine values.

 

[FvM]

yh = R*sin(theta) is the basic relation, as mentioned several times in previous posts, first in post #2. There are several methods to calculate sin() function, power series expansion was already mentioned, CORDIC is another option. Most methods imply an infinite number of operations for exact solution. CORDIC can be reduced to a single complex multiply for discrete angles.