A pair of conjugate semi-diameters of an ellipse

Geometry Level pending

Consider the ellipse described by

r ( t ) = C + f 1 cos ⁡ t + f 2 sin ⁡ t \mathbf{r}(t) = \mathbf{C} + \mathbf{f_1} \cos t + \mathbf{f_2} \sin t

The vectors f 1 , f 2 \mathbf{f_1} , \mathbf{f_2} can be shown to be conjugate semi-diameters of this ellipse. In this problem, we want to find another pair of vectors g 1 , g 2 \mathbf{g_1}, \mathbf{g_2} such that the same ellipse can be re-parameterized as,

r ( s ) = C + g 1 cos ⁡ s + g 2 sin ⁡ s \mathbf{r}(s) = \mathbf{C} + \mathbf{g_1} \cos s + \mathbf{g_2} \sin s

For this purpose, suppose we choose g 1 = cos ⁡ θ f 1 + sin ⁡ θ f 2 \mathbf{g_1} = \cos \theta \mathbf{f_1} + \sin \theta \mathbf{f_2} , for some θ \theta , then what must g 2 \mathbf{g_2} be ?

± sin ⁡ θ f 1 ± cos ⁡ θ f 2 \pm \sin \theta \mathbf{f_1} \pm \cos \theta \mathbf{f_2} ± cos ⁡ θ f 1 ± sin ⁡ θ f 2 \pm \cos \theta \mathbf{f_1} \pm \sin \theta \mathbf{f_2} ∓ cos ⁡ θ f 1 ± sin ⁡ θ f 2 \mp \cos \theta \mathbf{f_1} \pm \sin \theta \mathbf{f_2} ∓ sin ⁡ θ f 1 ± cos ⁡ θ f 2 \mp \sin \theta \mathbf{f_1} \pm \cos \theta \mathbf{f_2}

Hosam Hajjir by Hosam Hajjir , 56, Canada

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