Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

**ryos****Member**- Registered: 2005-08-04
- Posts: 394

Hey guys! Long time no post. I only come back when I've got a goofy problem. Sorry about that.

Anyway, I'll cut to the chase. A spiral is easy to describe in polar coordinates. It has the form r = aθ / 2π , where a is the amount by which r increases in each complete revolution. It's also not hard to map this equation to cartesian coordinates:

x = r*cos(θ)

y = r*sin(θ)

Here's the goofy problem: how do I find the (cartesian) slope of the line tangent to the curve at a given value of r and θ? I've tried implicit differentiation, like so:

dr/dθ = 1 / 2π

dx/dθ = (dr/dθ)*(-sin(θ))

dx/dθ = -sin(θ)/2π

dy/dθ = (dr/dθ)*(cos(θ))

= cos(θ)/2π

The slope of the line would then be dy/dx, and the angle said line makes with the x-axis would be atan( dy/dx ). Right? Am I getting this right? Because the angles produced from these expressions seem a bit off.

Thanks!

El que pega primero pega dos veces.

Offline

**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

thats how i work it out.

The Beginning Of All Things To End.

The End Of All Things To Come.

Offline

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

ryos wrote:

The slope of the line would then be dy/dx, and the angle said line makes with the x-axis would be atan( dy/dx ). Right?

The angle the tangent makes with the horizontal is tan[sup]−1[/sup](d*y*/d*x*).

*Last edited by JaneFairfax (2007-03-03 04:24:40)*

Offline

**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

luca-dd said: wrote:

*Last edited by John E. Franklin (2007-03-03 04:00:47)*

**igloo** **myrtilles** **fourmis**

Offline

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Good point, John! I completely missed that myself.

Fortunately I found this:

http://en.wikipedia.org/wiki/Polar_coor … l_calculus

Offline

**ryos****Member**- Registered: 2005-08-04
- Posts: 394

D'oh! I forgot the chain rule. Thanks Jane.

El que pega primero pega dos veces.

Offline

Pages: **1**