Find Contour Such that Only One Intersection Exists

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I'm trying to work out a visualization in Desmos, but I'm having trouble finding a closed form solution.



The problem is that I have a function



  1. $$(|x|^k + |y|^k)^frac1k = 5$$

that I would like to intersect only once with the following ellipse



  1. $$frac((x - 10)cos(-1) + (y - 10)sin(-1))^2a + frac((x - 10)cos(-1) - (y - 10)sin(-1))^2b = c.$$

How can find or set $c$ as a function of $k$ such that there is only one intersection?




algebra-precalculus geometry




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  • 1




    @Mason Equation (1) is just trying to model the Minowoski distances, so in some circumstances $k$ can take non-integer values from [0, inf]. The -1 is in fact radians. Equation (2) is just a general form for eclipse with -1 adding a skew / tilt/
    – Sentient
    Aug 1 at 2:12










  • Sorry, ellipse is the correct term.
    – Sentient
    Aug 1 at 2:17










  • @Mason Here's what I'm trying to work on: desmos.com/calculator/ohx8vtyipl. However, the green ellipse is current not a function of $k$. Ideally, when $k$ shifts, both the blue and green will intersect at only one point.
    – Sentient
    Aug 1 at 2:19










  • $a, b$ are both fixed so that we can find $c$ as a function of $k$
    – Sentient
    Aug 1 at 2:25






  • 1




    I kind of need a solution that generalizes to values of $k$ between [0, inf].
    – Sentient
    Aug 1 at 2:53














up vote
0
down vote

favorite












I'm trying to work out a visualization in Desmos, but I'm having trouble finding a closed form solution.



The problem is that I have a function



  1. $$(|x|^k + |y|^k)^frac1k = 5$$

that I would like to intersect only once with the following ellipse



  1. $$frac((x - 10)cos(-1) + (y - 10)sin(-1))^2a + frac((x - 10)cos(-1) - (y - 10)sin(-1))^2b = c.$$

How can find or set $c$ as a function of $k$ such that there is only one intersection?




algebra-precalculus geometry




share|cite|improve this question

















  • 1




    @Mason Equation (1) is just trying to model the Minowoski distances, so in some circumstances $k$ can take non-integer values from [0, inf]. The -1 is in fact radians. Equation (2) is just a general form for eclipse with -1 adding a skew / tilt/
    – Sentient
    Aug 1 at 2:12










  • Sorry, ellipse is the correct term.
    – Sentient
    Aug 1 at 2:17










  • @Mason Here's what I'm trying to work on: desmos.com/calculator/ohx8vtyipl. However, the green ellipse is current not a function of $k$. Ideally, when $k$ shifts, both the blue and green will intersect at only one point.
    – Sentient
    Aug 1 at 2:19










  • $a, b$ are both fixed so that we can find $c$ as a function of $k$
    – Sentient
    Aug 1 at 2:25






  • 1




    I kind of need a solution that generalizes to values of $k$ between [0, inf].
    – Sentient
    Aug 1 at 2:53












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I'm trying to work out a visualization in Desmos, but I'm having trouble finding a closed form solution.



The problem is that I have a function



  1. $$(|x|^k + |y|^k)^frac1k = 5$$

that I would like to intersect only once with the following ellipse



  1. $$frac((x - 10)cos(-1) + (y - 10)sin(-1))^2a + frac((x - 10)cos(-1) - (y - 10)sin(-1))^2b = c.$$

How can find or set $c$ as a function of $k$ such that there is only one intersection?




algebra-precalculus geometry




share|cite|improve this question













I'm trying to work out a visualization in Desmos, but I'm having trouble finding a closed form solution.



The problem is that I have a function



  1. $$(|x|^k + |y|^k)^frac1k = 5$$

that I would like to intersect only once with the following ellipse



  1. $$frac((x - 10)cos(-1) + (y - 10)sin(-1))^2a + frac((x - 10)cos(-1) - (y - 10)sin(-1))^2b = c.$$

How can find or set $c$ as a function of $k$ such that there is only one intersection?





algebra-precalculus geometry





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 1 at 2:20
























asked Aug 1 at 1:55









Sentient

439516




439516







  • 1




    @Mason Equation (1) is just trying to model the Minowoski distances, so in some circumstances $k$ can take non-integer values from [0, inf]. The -1 is in fact radians. Equation (2) is just a general form for eclipse with -1 adding a skew / tilt/
    – Sentient
    Aug 1 at 2:12










  • Sorry, ellipse is the correct term.
    – Sentient
    Aug 1 at 2:17










  • @Mason Here's what I'm trying to work on: desmos.com/calculator/ohx8vtyipl. However, the green ellipse is current not a function of $k$. Ideally, when $k$ shifts, both the blue and green will intersect at only one point.
    – Sentient
    Aug 1 at 2:19










  • $a, b$ are both fixed so that we can find $c$ as a function of $k$
    – Sentient
    Aug 1 at 2:25






  • 1




    I kind of need a solution that generalizes to values of $k$ between [0, inf].
    – Sentient
    Aug 1 at 2:53












  • 1




    @Mason Equation (1) is just trying to model the Minowoski distances, so in some circumstances $k$ can take non-integer values from [0, inf]. The -1 is in fact radians. Equation (2) is just a general form for eclipse with -1 adding a skew / tilt/
    – Sentient
    Aug 1 at 2:12










  • Sorry, ellipse is the correct term.
    – Sentient
    Aug 1 at 2:17










  • @Mason Here's what I'm trying to work on: desmos.com/calculator/ohx8vtyipl. However, the green ellipse is current not a function of $k$. Ideally, when $k$ shifts, both the blue and green will intersect at only one point.
    – Sentient
    Aug 1 at 2:19










  • $a, b$ are both fixed so that we can find $c$ as a function of $k$
    – Sentient
    Aug 1 at 2:25






  • 1




    I kind of need a solution that generalizes to values of $k$ between [0, inf].
    – Sentient
    Aug 1 at 2:53







1




1




@Mason Equation (1) is just trying to model the Minowoski distances, so in some circumstances $k$ can take non-integer values from [0, inf]. The -1 is in fact radians. Equation (2) is just a general form for eclipse with -1 adding a skew / tilt/
– Sentient
Aug 1 at 2:12




@Mason Equation (1) is just trying to model the Minowoski distances, so in some circumstances $k$ can take non-integer values from [0, inf]. The -1 is in fact radians. Equation (2) is just a general form for eclipse with -1 adding a skew / tilt/
– Sentient
Aug 1 at 2:12












Sorry, ellipse is the correct term.
– Sentient
Aug 1 at 2:17




Sorry, ellipse is the correct term.
– Sentient
Aug 1 at 2:17












@Mason Here's what I'm trying to work on: desmos.com/calculator/ohx8vtyipl. However, the green ellipse is current not a function of $k$. Ideally, when $k$ shifts, both the blue and green will intersect at only one point.
– Sentient
Aug 1 at 2:19




@Mason Here's what I'm trying to work on: desmos.com/calculator/ohx8vtyipl. However, the green ellipse is current not a function of $k$. Ideally, when $k$ shifts, both the blue and green will intersect at only one point.
– Sentient
Aug 1 at 2:19












$a, b$ are both fixed so that we can find $c$ as a function of $k$
– Sentient
Aug 1 at 2:25




$a, b$ are both fixed so that we can find $c$ as a function of $k$
– Sentient
Aug 1 at 2:25




1




1




I kind of need a solution that generalizes to values of $k$ between [0, inf].
– Sentient
Aug 1 at 2:53




I kind of need a solution that generalizes to values of $k$ between [0, inf].
– Sentient
Aug 1 at 2:53















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