Intrinsic coordinates
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Intrinsic coordinates Questions Exercise 1A - Problems on finding intrinsic coordinates
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(1) Let the parametric equation [Equation goes here - download the original to see it.] . Prove that [Equation goes here - download the original to see it.] . Deduce that [Equation goes here - download the original to see it.] , and hence find the intrinsic equation. (2) For the curve [Equation goes here - download the original to see it.] prove that [Equation goes here - download the original to see it.] and that [Equation goes here - download the original to see it.] . Hence find the intrinsic equation. (3) For the curve [Equation goes here - download the original to see it.] Prove that [Equation goes here - download the original to see it.] and that the intrinsic equation is [Equation goes here - download the original to see it.] when [Equation goes here - download the original to see it. ] and [Equation goes here - download the original to see it.] (4) Find the intrinsic equation of the astroid [Equation goes here - download the original to see it.] Prove that [Equation goes here - download the original to see it.] (5) Let YOX be a rectangular system of coordinates. The tractrix which lies on YOX is the curve [Equation goes here - download the original to see it.] whose tangent which intersects the x-axis is constant in length and equal to a (see the diagram). [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] (i) Show that the parametric equation of the tractrix is [Equation goes here - download the original to see it.] which passes through [Equation goes here - download the original to see it.] (ii) Prove that [Equation goes here - download the original to see it.] (iii) Prove that the intrinsic equation is [Equation goes here - download the original to see it.] (6) If the parametric equation of a curve is [Equation goes here - download the original to see it.] find the intrinsic equation. (7) Given that [Equation goes here - download the original to see it.] and that [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] when [Equation goes here - download the original to see it.], find the parametric equation of the curve. (8) Find the intrinsic equation of the logarithmical spiral [Equation goes here - download the original to see it.] Solutions Exercise 1A (1) [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] From this, we have that [Equation goes here - download the original to see it.] Therefore, [Equation goes here - download the original to see it.] Since [Equation goes here - download the original to see it.] , we get that [Equation goes here - download the original to see it.] From this [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] (1) We know that [Equation goes here - download the original to see it.] But [Equation goes here - download the original to see it.] , which is obtained from the partial integration. From this, we get that [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] , i.e [Equation goes here - download the original to see it.] Therefore, from (1), we get that [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] , with when [Equation goes here - download the original to see it.]. (2) We have [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] . But [Equation goes here - download the original to see it.] From these facts, we obtain that [Equation goes here - download the original to see it.] From the equation [Equation goes here - download the original to see it.] , we get [Equation goes here - download the original to see it.] , i.e [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] , with [Equation goes here - download the original to see it.] when [Equation goes here - download the original to see it.].
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Contents of
Intrinsic coordinates
1 Intrinsic coordinates Questions Exercise 1A - Problems on finding intrinsic coordinates
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