In algebraic geometry, the first polar, or simply polar of an algebraic plane curve C of degree n with respect to a point Q is an algebraic curve of degree n−1 which contains every point of C whose tangent line passes through Q. It is used to investigate the relationship between the curve and its dual, for example in the derivation of the Plücker formulas.
Then ΔQf is a homogeneous polynomial of degree n−1 and ΔQf(x, y, z) = 0 defines a curve of degree n−1 called the first polar of C with respect of Q.
If P=(p, q, r) is a non-singular point on the curve C then the equation of the tangent at P is
In particular, P is on the intersection of C and its first polar with respect to Q if and only if Q is on the tangent to C at P. Note also that for a double point of C, the partial derivatives of f are all 0 so the first polar contains these points as well.
The class of C may be defined as the number of tangents that may be drawn to C from a point not on C (counting multiplicities and including imaginary tangents). Each of these tangents touches C at one of the points of intersection of C and the first polar, and by Bézout's theorem theorem there are at most n(n−1) of these. This puts an upper bound of n(n−1) on the class of a curve of degree n. The class may be computed exactly by counting the number and type of singular points on C (see Plücker formula).
A polar curve is a graph which contrasts the sink rate of an aircraft (typically a glider) with its horizontal speed.
Knowing the best speed to fly is important in exploiting the performance of a glider. Two of the key measures of a glider’s performance are its minimum sink rate and its best glide ratio, also known as the best 'glide angle'. These occur at different speeds. Knowing these speeds is important for efficient cross-country flying. In still air the polar curve shows that flying at the minimum sink speed enables the pilot to stay airborne for as long as possible and to climb as quickly as possible, but at this speed the glider will not travel as far as if it flew at the speed for the best glide. When in sinking air, the polar curve shows that best speed to fly depends on the rate that the air is descending. Using Paul MacCready's theory, the optimal speed to fly for best cross country speed may often be considerably in excess of the speed for the best glide angle to get out of the sinking air as quickly as possible.
Why do I trust you ten years
Standing on the line
only to find out
I'll be laid off in July
Someday I swear
I'll buy and sell your soul
But in the meantime
I'll keep on shoveling coal
[Chorus]
One on the rack
And the next is coming
My hands are calloused and I
I can't take the heat
This place is like an oven
Why do I try
If I leave here without pension
And I go to the grave
My life will be forced to love another
For the rent that will he will pay
Did you ever bother to think of the repercussions
Of dealing with a guy like me
It's not like you pay me minimum wage
But now you're taking my dignity
[Chorus]
One on the rack
And the next is coming
My hands are calloused and I
I can't take the heat
This place is like an oven
Why do I try
[Chorus]
One on the rack
And the next is coming
My hands are calloused and I
I can't take the heat
This place is like an oven