Sign of the scalar product of vectors centered at the foci in an ellipsoid
$begingroup$
Consider an ellipsoid for the vector $vec{x}$ with equation
begin{equation}
sqrt{ lVertvec{x}rVert^2+a^2}+sqrt{lVertvec{x}+vec{c}rVert^2+a^2}=d
end{equation}
and parameters $a$, $d$ and $vec{c}$.
Is it true, and how would you proove, the following statement:
The scalar product of $2vec{x}+vec{c}$ and $vec{x}$ is always positive.
Specifically, $2vec{x}+vec{c}$ is the sum of the two vectors $vec{x}$ and $vec{x}+vec{c}$ starting at the foci, ending and pointing to the point $vec{x}$ on the ellipsoid.
geometry conic-sections
asked Dec 21 '18 at 14:59
TanatofobicoTanatofobico
425
$endgroup$
add a comment |
$begingroup$
Consider an ellipsoid for the vector $vec{x}$ with equation
begin{equation}
sqrt{ lVertvec{x}rVert^2+a^2}+sqrt{lVertvec{x}+vec{c}rVert^2+a^2}=d
end{equation}
and parameters $a$, $d$ and $vec{c}$.
Is it true, and how would you proove, the following statement:
The scalar product of $2vec{x}+vec{c}$ and $vec{x}$ is always positive.
Specifically, $2vec{x}+vec{c}$ is the sum of the two vectors $vec{x}$ and $vec{x}+vec{c}$ starting at the foci, ending and pointing to the point $vec{x}$ on the ellipsoid.
geometry conic-sections
asked Dec 21 '18 at 14:59
TanatofobicoTanatofobico
425
$endgroup$
$begingroup$
Are you sure the equation is correct?
$endgroup$
– Aretino
Dec 21 '18 at 15:55
$begingroup$
maybe you are right, the same-major axis is not long enough. Would the veridicity of the statement change if I just chose an arbitrary major axis-length?
$endgroup$
– Tanatofobico
Dec 21 '18 at 16:29
$begingroup$
That scalar product is positive if the angle between vectors $vec x$ and $vec x+vec c$ is always acute. But that is not the case if the ellipsoid is stretched enough.
$endgroup$
– Aretino
Dec 21 '18 at 16:35
$begingroup$
Indeed, but I was asking between $2vec{x}+vec{c}$ and $vec{x}$.
$endgroup$
– Tanatofobico
Dec 21 '18 at 19:24
$begingroup$
The answer to the statement is no, anyway. One can proove as an exercise of constrained optimization that if the eccentricity of the ellipse is higher than $frac{3}{4}$ then there always is a point where $(2vec{x}+vec{c})cdotvec{x}$ is negative.
$endgroup$
– Tanatofobico
Dec 22 '18 at 13:07
add a comment |
$begingroup$
Consider an ellipsoid for the vector $vec{x}$ with equation
begin{equation}
sqrt{ lVertvec{x}rVert^2+a^2}+sqrt{lVertvec{x}+vec{c}rVert^2+a^2}=d
end{equation}
and parameters $a$, $d$ and $vec{c}$.
Is it true, and how would you proove, the following statement:
The scalar product of $2vec{x}+vec{c}$ and $vec{x}$ is always positive.
Specifically, $2vec{x}+vec{c}$ is the sum of the two vectors $vec{x}$ and $vec{x}+vec{c}$ starting at the foci, ending and pointing to the point $vec{x}$ on the ellipsoid.
geometry conic-sections
asked Dec 21 '18 at 14:59
TanatofobicoTanatofobico
425
$endgroup$
Consider an ellipsoid for the vector $vec{x}$ with equation
begin{equation}
sqrt{ lVertvec{x}rVert^2+a^2}+sqrt{lVertvec{x}+vec{c}rVert^2+a^2}=d
end{equation}
and parameters $a$, $d$ and $vec{c}$.
Is it true, and how would you proove, the following statement:
The scalar product of $2vec{x}+vec{c}$ and $vec{x}$ is always positive.
Specifically, $2vec{x}+vec{c}$ is the sum of the two vectors $vec{x}$ and $vec{x}+vec{c}$ starting at the foci, ending and pointing to the point $vec{x}$ on the ellipsoid.
geometry conic-sections
geometry conic-sections
asked Dec 21 '18 at 14:59
TanatofobicoTanatofobico
425
asked Dec 21 '18 at 14:59
TanatofobicoTanatofobico
425
edited Dec 21 '18 at 16:29
Tanatofobico
asked Dec 21 '18 at 14:59
TanatofobicoTanatofobico
425
asked Dec 21 '18 at 14:59
TanatofobicoTanatofobico
425
asked Dec 21 '18 at 14:59
TanatofobicoTanatofobico
425
425
$begingroup$
Are you sure the equation is correct?
$endgroup$
– Aretino
Dec 21 '18 at 15:55
$begingroup$
maybe you are right, the same-major axis is not long enough. Would the veridicity of the statement change if I just chose an arbitrary major axis-length?
$endgroup$
– Tanatofobico
Dec 21 '18 at 16:29
$begingroup$
That scalar product is positive if the angle between vectors $vec x$ and $vec x+vec c$ is always acute. But that is not the case if the ellipsoid is stretched enough.
$endgroup$
– Aretino
Dec 21 '18 at 16:35
$begingroup$
Indeed, but I was asking between $2vec{x}+vec{c}$ and $vec{x}$.
$endgroup$
– Tanatofobico
Dec 21 '18 at 19:24
$begingroup$
The answer to the statement is no, anyway. One can proove as an exercise of constrained optimization that if the eccentricity of the ellipse is higher than $frac{3}{4}$ then there always is a point where $(2vec{x}+vec{c})cdotvec{x}$ is negative.
$endgroup$
– Tanatofobico
Dec 22 '18 at 13:07
add a comment |
$begingroup$
Are you sure the equation is correct?
$endgroup$
– Aretino
Dec 21 '18 at 15:55
$begingroup$
maybe you are right, the same-major axis is not long enough. Would the veridicity of the statement change if I just chose an arbitrary major axis-length?
$endgroup$
– Tanatofobico
Dec 21 '18 at 16:29
$begingroup$
That scalar product is positive if the angle between vectors $vec x$ and $vec x+vec c$ is always acute. But that is not the case if the ellipsoid is stretched enough.
$endgroup$
– Aretino
Dec 21 '18 at 16:35
$begingroup$
Indeed, but I was asking between $2vec{x}+vec{c}$ and $vec{x}$.
$endgroup$
– Tanatofobico
Dec 21 '18 at 19:24
$begingroup$
The answer to the statement is no, anyway. One can proove as an exercise of constrained optimization that if the eccentricity of the ellipse is higher than $frac{3}{4}$ then there always is a point where $(2vec{x}+vec{c})cdotvec{x}$ is negative.
$endgroup$
– Tanatofobico
Dec 22 '18 at 13:07
$begingroup$
Are you sure the equation is correct?
$endgroup$
– Aretino
Dec 21 '18 at 15:55
$begingroup$
Are you sure the equation is correct?
$endgroup$
– Aretino
Dec 21 '18 at 15:55
$begingroup$
maybe you are right, the same-major axis is not long enough. Would the veridicity of the statement change if I just chose an arbitrary major axis-length?
$endgroup$
– Tanatofobico
Dec 21 '18 at 16:29
$begingroup$
maybe you are right, the same-major axis is not long enough. Would the veridicity of the statement change if I just chose an arbitrary major axis-length?
$endgroup$
– Tanatofobico
Dec 21 '18 at 16:29
$begingroup$
That scalar product is positive if the angle between vectors $vec x$ and $vec x+vec c$ is always acute. But that is not the case if the ellipsoid is stretched enough.
$endgroup$
– Aretino
Dec 21 '18 at 16:35
$begingroup$
That scalar product is positive if the angle between vectors $vec x$ and $vec x+vec c$ is always acute. But that is not the case if the ellipsoid is stretched enough.
$endgroup$
– Aretino
Dec 21 '18 at 16:35
$begingroup$
Indeed, but I was asking between $2vec{x}+vec{c}$ and $vec{x}$.
$endgroup$
– Tanatofobico
Dec 21 '18 at 19:24
$begingroup$
Indeed, but I was asking between $2vec{x}+vec{c}$ and $vec{x}$.
$endgroup$
– Tanatofobico
Dec 21 '18 at 19:24
$begingroup$
The answer to the statement is no, anyway. One can proove as an exercise of constrained optimization that if the eccentricity of the ellipse is higher than $frac{3}{4}$ then there always is a point where $(2vec{x}+vec{c})cdotvec{x}$ is negative.
$endgroup$
– Tanatofobico
Dec 22 '18 at 13:07
$begingroup$
The answer to the statement is no, anyway. One can proove as an exercise of constrained optimization that if the eccentricity of the ellipse is higher than $frac{3}{4}$ then there always is a point where $(2vec{x}+vec{c})cdotvec{x}$ is negative.
$endgroup$
– Tanatofobico
Dec 22 '18 at 13:07
add a comment |
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$begingroup$
Are you sure the equation is correct?
$endgroup$
– Aretino
Dec 21 '18 at 15:55
$begingroup$
maybe you are right, the same-major axis is not long enough. Would the veridicity of the statement change if I just chose an arbitrary major axis-length?
$endgroup$
– Tanatofobico
Dec 21 '18 at 16:29
$begingroup$
That scalar product is positive if the angle between vectors $vec x$ and $vec x+vec c$ is always acute. But that is not the case if the ellipsoid is stretched enough.
$endgroup$
– Aretino
Dec 21 '18 at 16:35
$begingroup$
Indeed, but I was asking between $2vec{x}+vec{c}$ and $vec{x}$.
$endgroup$
– Tanatofobico
Dec 21 '18 at 19:24
$begingroup$
The answer to the statement is no, anyway. One can proove as an exercise of constrained optimization that if the eccentricity of the ellipse is higher than $frac{3}{4}$ then there always is a point where $(2vec{x}+vec{c})cdotvec{x}$ is negative.
$endgroup$
– Tanatofobico
Dec 22 '18 at 13:07