Questions tagged [analytic-geometry]
Questions on the use of algebraic techniques for proving geometric facts. Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. It is concerned with defining and representing geometrical shapes in a numerical way.
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Parabolic Arc as a Product of 2 Latus Rectum Segments
I have been searching for any published work on this definition of a parabolic arc. The latus rectum of a parabola is a defining chord drawn parallel to the directrix and passes thru the focus.
In the ...
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Coordinate-free proof of the identity for two times cross product operator
Consider an oriented three-dimensional Euclidean space $V$. Let $[a]$ be the operator of cross product by the vector $a$: $[a] b = a \times b$. It is easy to check that
$$[a]^2 = a \otimes \flat a - \...
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Solving for a system of two equations of a plane or the intersection of two planes when bounds are given instead of zero in the right hand side
I have read this question to find the parametric form for the line formed by the intersection of two planes.
But I have two equations that have bounds instead of 0 on the right hand side:
$$B_{min} &...
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Restricting maximum curvature of cubic bezier curve
Is there any way to reasonably restrict control points of cubic bezier curve so it's oscilating circle will never have radius smaller than r?
Bezier curve with it's ...
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We can't write $\mathbb{R}P^{n}=\mathbb{R}P\times \cdots \times \mathbb{R}P$
please help me to find answer this question,
Why can't we write,
$\mathbb{R}P^{n}=\mathbb{R}P\times \cdots \times \mathbb{R}P$
if $\mathbb{R}P^{n}$ is real projective space?
I know (in geometry) $\...
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What is the answer to the following ant puzzle?
There are n points marked on the circumference of a circle, the circumference having length 1 m. There are n ants on these n points initially. At some point of time the n ants start moving, some ...
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Line integral of closed and otherwise non-intersecting curves on a cylindric surface
In my multi variable/vector calculus textbook I encountered a problem which led to more questions than answers when i solved it. Let me start by stateing the problem:
The problem
Let $\gamma$ be a ...
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The shortest sum of distances from two points to point is $Oxy$ plane
I have next problem:
Find the point $C$ on the $Oxy$ plane, the sum of the distances from it to the points $A(-7,3,5)$ and $B(2,-2,3)$ would be the smallest.
There is my progress:
Let us assume that ...
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Solving a circle equation using tangent property
Let $P$ be the point in $XY$ plane and $P'$ is a point such that $OP.OP'=r^2$, where $O,P,P'$ are collinear and $O$ divides $P$ and $P'$ externally and $O$ is the origin.
If the point $P$ lies on the ...
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Determine the type of triangle constructed on the polynomials.
Determine the type of triangle constructed on the polynomials:
2-t+5t² , 3t²+2t+1 ,
if the scalar product is defined as follows:
(f,g) = a₀b₀ + 2a₁b₁ + 3a₂b₂
I just don't understand what 'triangle ...
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Finding the dimensions of a box (cuboid) given a hexagon filled in to look like the box
Suppose I have hexagons that like the ones below but I know the area and the points of each hexagon that represent a cuboid of dimensions g,h,d. How can I find the values for g, h, and d? Any pointers ...
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Determine the type of triangle constructed on the polynomials [closed]
Determine the type of triangle constructed on the polynomials:
2-t+5t² , 3t²+2t+1 ,
if the scalar product is defined
as follows:
(f,g) = a₀b₀ + 2a₁b₁ + 3a₂b₂
I just don't understand what 'triangle ...
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Area Between Hyperbola and its Tangent Lines
Draw tangent lines from point $\left(\frac{3}{5},0\right)$ on a hyperbola $x^2-y^2=1$ and then find the area defined by these tangent lines and a hyperbola.
Using some basic analytic geometry we can ...
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Indetermination in biarc construction
I'm following this friendly article in order to compute a biarc in $\mathbb{R}^2$. The author writes
Choosing a value for $d_1$ plays a big role in the shape of the biarc. Negative values will make ...
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Defining a mapping which rotates a plane
Let $n = (n_1, n_2, n_3) \in \mathbb{R}^3 \setminus \left\{0\right\}$. I want to define a linear map $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, $f(x)=Ax$, which maps the xy-plane $z = 0$ to the plane
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How do I find the equation of a circle given two tangents and a line which contains its centre but no points
How do I find the equation of the circle which touches both the x-axis (that is y = 0 is a tangent) and the line 4x - 3y + 4 = 0, find its centre lies in the first quadrant and on the line x - y - 1 = ...
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Largest number of equidistant points in $\mathbb{R}^n$ [closed]
Let $X$ be contained in $\mathbb{R}^2$ such that the euclidian metric restricted to $X$ equals the discrete metric. Prove that $X$ has at most $3$ elements. What if $X$ is contained in $\mathbb{R}^3$? ...
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does angle between tangents of two surfcaces at the point of intersection is same as angle between their normals at that point.
I have to find the value of a and b such that the curves:
$ax^2-byz=(a+2)x$ and $4x^2y+z^3=4$ are orthogonal at point (1,-1,2), Now we know that angle between curves is the angle between their ...
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Solving for DE in an Acute Triangle with Circumcenter and Euler Line [closed]
Define acute △ABC with circumcenter O. The circumcircle of △ABO meets segment
BC at D ≠ B, segment AC at F ≠ A, and the Euler line of △ABC at P ≠ O. The circumcircle of △ACO meets segment BC at E ≠ C. ...
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If a segment has an endpoint at the tangent point of a hyperplane to a hypersphere and no other intersection then it must be at the other half-space
I am trying to prove the following thing that in 2 or 3 dimensions seems intuitive:
In $\mathbb{R}^n$, let $\varphi$ be a hyperplane passing by the point $Q$ and let $\mathbb{S}$ be a hypersphere, ...
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Elegantly solvable game always ends in finite number of steps
The problem I'll describe below looks very difficult to me when trying to approach it with common mathematical processes (I'm not very skilled), but I've noted a good relation with physics which ...
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Randomly generate points uniformly on an ellipsoid in general position
I want to uniformly generate points at random on the boundary of an ellipsoid in any dimension.
The method given by @elhuhdron here is very nice and it can be straightforwardly generalized to any ...
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Where should a point be located at a line so that the difference of distances of two other points lying on same side of the line from it is maximised?
There is a line $L$ and two points $A$ and $B$ not lying on it. The points are on the same side of line. A point $C$ is located on the line $L$ such that $|AC - BC|$ is maximised. Determine $C$.
This ...
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Vector triple product as a condition of intersection of two lines
Let's say there are two lines of the form:
$$p_0=\frac{x-x_0}{l_0}=\frac{y-y_0}{m_0}=\frac{z-z_0}{n_0}$$
$$p_1=\frac{x-x_1}{l_1}=\frac{y-y_1}{m_1}=\frac{z-z_1}{n_1}$$
For $p_0$ and $p_1$ to intersect ...
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From any point $P$ on the line $5x+4y=20$ tangents are drawn to the circle $x^2+y^2=4$ then find the locus of the circum-centre of the $\triangle PQR$
From any point $P$ on the line $5x+4y=20$ tangents are drawn to the circle $x^2+y^2=4$ meeting at $Q$ and $R$, then find the locus of the circum-centre of $\triangle PQR$.
My approach is as follow, if ...
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Height of a cone given its radius
this is my first time posting. I have tried this solving this question but am unable to find a solution. We are supposed to use integration. Please help me understand what I am doing wrong? Thank you.
...
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Is it trivial that the directrices of an ellipse are perpendicular to its focal axis?
All the proofs I've so far seen of the equation of the directrix of an ellipse based on its eccentricity (and also those which aren't) assume that the directrices are perpendicular to the focal axis. ...
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Equation of a line passing through a point and perpendicular to another line
Find the equation of a line $n$ passing through point $T(2,3,1)$ which is also perpendicular to a line given by equation $$p:\frac{x+1}{2}=\frac{y}{-1}=\frac{z+2}{1}$$
I did this using two methods ...
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Triangle inside parabola/projectile
My doubt is that can a right triangle be constructed in a projectile motion by keeping the projectiles's range as hypotenuse of the right triangle and vertex opposite to its hypotenuse on the ...
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Rational parameterization of the unit circle using a bounded interval
It is well known that a part of the unit circle can be rationally parameterized by
$$x=\frac{1-t^2}{1+t^2},\,y=\frac{2t}{1+t^2}$$
where $-\infty \lt t \lt \infty$. However, there's no $t\in\mathbb{R}$ ...
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The tangent at (12,6) to a parabola intersects its directrix at (-1,2). The focus of the parabola lies on x-axis. number of such parabolas is
I tried to proceed by assuming the focus to be $(a,0)$. Then I found the equation of tangent which came out to be $4x-13y+30=0$.
I used the property of a parabola that the image of its focus about a ...
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Ray-cone and ray-cylinder intersections
I am writing a simple raytracer, and I want to be able to raytrace capped vertical cone and cylinder primitives. I need to be able to find intersection points and normals for use in lighting and ...
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Locus of a point from which tangents drawn to a fixed circle are inclined at a constant angle $\theta$ and area bounded by it
The question is given above. I'm able to get the locus, however, I'm unable to comment on the area bounded by it.
We can get the locus using the relation $\tan\theta=\frac{2LR}{L^{2}-R^{2}}$ where $...
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If diagonal points of a square are sliding on coordinate axes, locus of other two points
The full question is shown above. We have to try and write coordinates of A and C in terms of a single variable which we can later eliminate, but I'm unable to accomplish this.
I tried using ...
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Find the point in $S=\{(x, y)| x, y \text{ are positive integers }\}$ which have the least possible distance sum from the points $(0,12)$ and $(8,0).$
Let $S=\{(x, y)| x, y \text{ are positive integers }\}$ viewed as a subset of the plane. For every point $P$ in $S,$ let $d_P$ denote the sum of the distances from $P$ to the point $(8,0)$ and the ...
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What kind of geometric curve is this?
I am working on an optimal control problem,more specifically on minimum-time control under input constraint $|u(t)| \leq 1$, and I need to figure out the commutation curve of a bang-bang control (so $...
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Proving the locus of $P_k$ is an ellipse.
I was recently working on the following problem functioning within an overlap of complex numbers and coordinate geometry:
Let $z$ be a complex number $a + ib$ (where $a > b > 0$), and $α_k$ ($0 &...
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Why do the sign conventions in cartesian geometry work the way they do?
Usually in a Cartesian form of derivation in math and physics, I have seen that a particular formula is derived for simplicity by taking concerned points say in the first quadrant. Few examples are ...
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Proof that the midpoint of two given points are collinear
So I wanted to prove in an alternative way that the midpoint of two given points are collinear. The first way was as follows : Let $A,B,C\in \mathbb{R}^2$ be given by $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $...
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Show that the graph of the equation $x^{3}+3 x^{2} y+3 x y^{2}+y^{3}-x^{2}+y^{2}=0$ is a union of line and a parabola.
Show that the graph of the equation $$x^{3}+3 x^{2} y+3 x y^{2}+y^{3}-x^{2}+y^{2}=0$$ is a union of line and a parabola.
This seems a pretty easy question, but the answer I got is not matching with ...
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a straight line divides the ellipse into two parts, find the distance between the centers of mass of its parts
$3x+3y-9=0$ divides the ellipse $\frac{x^2}{25}+\frac{y^2}{4} = 1$ into two parts. Find the distance between the centers of mass of its parts
I realized that I need to make several double integrals, ...
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Mirror reflection and integrating rays coming back at focal point
Let a conic section in 2D :
$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$
For a circle with radius $R, B=D=E=0$ and $F=R2, A=B=1$.
The conic section may be assumed a mirror.
Let us assume some optical rays originating ...
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Norms of vectors in a equilateral triangle
Given the equilateral triangle $ABC$, I know $A(0,0,0)$ and I'm trying to find the coordinates of $B$ and $D$.
I know
$$AB\cdot AC=8$$
and the normalized vectors associated to them are
$$AB_0=(1,0,0)$$...
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Line through two points of parallelepiped
Given the parallelepiped
I need to find the equation of the line through the points $A$ and $F$. I know the line which contains $DG$ has equation given by
$$X=t(1,2,-3),\quad t\in\mathbb{R}$$
and $F(...
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Distance of centre of circle passing through points of contact of direct common tangents of two circles from the tangents.
We have this situation:
where E is the center of the circle passing through the points of contact of the direct common tangents.
A teacher claims that $x=\frac{r_1 +r_2}{2}$ and that it's true for ...
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How to find center of an arc given a geographical start point, end point and arc angle?
I am working in a way of drawing transitions in some kind of "Flight Plan" between two "waypoints" using the heading at the beginning of the arc and at the end of an arc. Given an ...
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Stereographic projection maps lines and circles on the plane into circles in the sphere
From pages $19$ and $20$ of Ahlfors' Complex Analysis:
Why is the converse true? Why does any line or circle in the plane mapped into a circle in the sphere?
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How to find a set of linear inequalities from the vertices of a $d$-dimensional convex polytope?
Let $S = \{x_0, \dots, x_n\} \subseteq (\mathbb{R}^+)^{d}$ be the set of vertices of a convex $d$-dimensional convex polytope ($d \geq 2$).
I am interested in finding a set of linear inequalities such ...
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Signed distance function about a circular arc
I have a rectangle of 1.3 mm * 0.2 mm, take the origin to the lower left corner of the rectangle. Say, I have a circular arc defined by the (0.3,0),(0.2732,0.1) & (0.3,0.2). It has a radius of 0.2 ...
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Why does the following equation sometimes describe a circle and sometimes a line?
From page $19$ of Ahlfors' Complex Analysis:
"[...] this equation takes the form [...]
$$(\alpha_0-\alpha_3)(x^2+y^2)-2\alpha_1x-2\alpha_2y+\alpha_0+\alpha_3 = 0.$$
For $\alpha_0\ne \alpha_3$ ...
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