How do you determine the coordinates on a cartesian grid from point A to point B along a curve?

Question

If given the radius in feet and the arclength in feet, how do I compute the change in coordinates from point A to point B on a two dimensional graph? This is a common occurence in land descriptions. I believe it is called a simple curve. I understand how to calculate the chord, but I don't know how figure the difference in coordinates between point A (e.g. x = 1000 and y = 900) and point B (x = ?, y = ?). Seems like there's a way to reduce it to a right triangle, but I can't see how.

Answer 1

It's not possible to compute the coordinates of B simply from the radius and arclength of a circular arc from A to B. We must also know the orientation of A with respect to the centre of the circle. Otherwise, we have a fixed arc shape with one end tied to A, and we can sweep it around A to generate a locus of possibilities for B.

Call the centre of the circle O and suppose that A is at an angle θ from O (going anticlockwise from the positive x axis, as is usual in mathematics). If the radius is r and the arc length is L, the angle between A and B is given by φ = L / r (in radians) and so B is at an angle (θ + φ) from O - assuming the curve is also in an anticlockwise direction; otherwise, replace (θ + φ) by (θ - φ) everywhere below.

Then, if the coordinates of A are (m, n), the coordinates of B are
(m + r cos (θ+φ) - r cos θ, n + r sin (θ+φ) - r sin θ).

Answer 2

a) f ' (x) = -4x^3 + 4x + 3. At x = a million, f ' (x) = 3. f (a million) = 4. So, you're stunning, y = 3x + a million. b) f " (x) = - 12x^2 + 4. jointly as is f " (x) >=0? this is the respond because of the fact this is a mandatory and sufficient challenge to teach that f " (x) >=0 on a collection is equivalent to f (x) being convex on the set.