Questions on the use of algebraic techniques to prove geometric theorems.
61
votes
18answers
6k views
How to check if a point is inside a rectangle?
There is a point (x,y), and a rectangle a(x1,y1),b(x2,y2),c(x3,y3),d(x4,y4), how can one check if the point inside the ...
16
votes
6answers
3k views
Is there an equation to describe regular polygons?
For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides ...
3
votes
1answer
703 views
Analogue of spherical coordinates in $n$-dimensions
What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by
...
8
votes
1answer
287 views
Navigating though the surface of a hypersphere in a computer game
People in StackOverflow seems not so into this theme, so I thought I could have better luck in here.
I had the idea of an spaceship game where the world is confined in the surface of an 4-D ...
4
votes
2answers
246 views
Applied ODEs in trajectory problem
I'm having a hard time solving this problem:
Let there be a town $A$ in a shore of a river. Let $x=0$ be the shore. Let $(0,0)$ be the location of the town. Let $B$ be another town, in the ...
4
votes
4answers
780 views
How to find the distance between a point and line joining two points on a sphere?
How do I calculate the distance between the line joining the two points on a spherical surface and another point on same surface? I have illustrated my problem in the image below.
In the above ...
3
votes
5answers
401 views
Help understanding cross-product
I am trying to calculate the intersection point (if any) of two line segments for a 2D computer game. I am trying to use this method, but I want to make sure I understand what is going on as I do it. ...
3
votes
2answers
147 views
Why can any affine transformaton be constructed from a sequence of rotations, translations, and scalings?
A book on CG says:
... we can construct any affine transformation from a sequence of rotations, translations, and scalings.
But I don't know how to prove it.
Even in a particular case, I found ...
2
votes
4answers
681 views
Proving two lines trisects a line
A question from my vector calculus assignment. Geometry, anything visual, is by far my weakest area. I've been literally staring at this question for hours in frustrations and I give up (and I do mean ...
7
votes
2answers
3k views
Finding the intersecting points on two circles
Given 2 circles on a plane, how do you calculate the intersecting points?
In this example I can do the calculation using the equilateral triangles that are described by the intersection and centres ...
5
votes
3answers
438 views
A simple(?) Analytical Geometry Question (Ellipse) my teacher can't solve
Here's the story:
I am a high school student who absolutely loves math. So I took a university level mathematics course that is renowned throughout our school for being extremely rigorous and tough. ...
2
votes
3answers
186 views
How to obtain Line Equation of the form $ax + by + c = 0$
I'm trying to check if a line hits a rectangle, and for that, I found this nice solution:
Line triangle intersection
The problem is that, having forgot almost all I ever knew about math, I don't ...
1
vote
1answer
156 views
shortest distance between two points on $S^2$
Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$
Length of a curve in $3D$ is ...
1
vote
1answer
198 views
How to find the intersection of the area of multiple triangles
I have a couple of questions regarding finding the intersection of triangles. I have a system of 16 projectors that all have slightly different color gamuts. The color gamuts are represented by a ...
1
vote
1answer
167 views
Exercise review: perpendicular-to-plane line
Please, can you check the following execution is correct:
Problem text
I have a plane in affine space in $\Bbb R^4$ described by two following equations:
\begin{Bmatrix}3x+y-z-q +1=0\\ ...
0
votes
2answers
336 views
Calculate Spherical Distance between points
I have googled this and not come up with an answer yet, but basically, I'm trying to find out the distance between each point or vertice on a sphere (all points are evenly spaced).
I already know this ...
0
votes
1answer
54 views
Can any smooth planar curve which is closed, be a base for a 3 dimensional cone?
A cone in 3 dimensions has a vertex and a base. The contour of the base is a circle which is a smooth closed planar curve. Can there be a more general cone which can have any smooth closed planar ...
-3
votes
1answer
44 views
Finding Coordinates of sphere
Find the coordinates of one point on the intersection of the sphere $$(x-1)^2+(y-2)^2+(z-4)^2=25$$ and the plane $$z=4.$$
Supply evidence to support your answer.
3
votes
3answers
642 views
Find a point on a line segment, located at a distance $d$ from one endpoint
Given points $A$ and $C$ in the plane, how do I find the point $B$ on the line segment between $A$ and $C$ that is located at a distance $d$ from $A$?
Example:
$$A = (0,3), \qquad C = (3,0), \qquad ...
9
votes
3answers
648 views
The vertices of an equilateral triangle are shrinking towards each other
For an equilateral triangle ABC of side $a$ vertex A is always moving in the direction of vertex B, which is always moving the direction of vertex C, which is always moving in the direction of vertex ...
6
votes
3answers
347 views
Parametric form of an ellipse given by $ax^2 + by^2 + cxy = d$
If $c = 0$, the parametric form is obviously $x = \sqrt{\frac{d}{a}} \cos(t), y = \sqrt{\frac{d}{b}} \sin(t)$.
When $c \neq 0$ the sine and cosine should be phase shifted from each other. How do I ...
1
vote
2answers
86 views
Max and min value of $7x+8y$ in a given half-plane limited by straight lines?
So, there are four inequalities:
$$\begin{eqnarray*}
y &\geq &-3x+15; \\
y &\leq &-11/3x+56/3; \\
x &\geq &0; \\
y &\geq &0.
\end{eqnarray*}$$
If we draw all those ...
5
votes
1answer
519 views
Formula for curve parallel to a parabola
I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
4
votes
1answer
162 views
what are some isometries of S^2 without fixed points?
Spherical geometry question involving isometries.
Particularly looking for isometries with no fixed points.
4
votes
3answers
675 views
Find a plane perpendicular to a plane passing by point
In $\mathbb R^4$ I have: $$\pi: \begin{cases} x+y-z+q+1=0 \\ 2x+3y+z-3q=0\end{cases}$$
I have to find $\pi' \bot$ $ \pi $ and passing by $P=(0,1,0,1)$. How can I do that? Thanks a lot!
4
votes
4answers
587 views
Find the centre of a circle passing through a known point and tangential to two known lines
I am trying to find the centre and radius of a circle passing through a known point, and that is also tangential to two known lines.
The only knowns are:
Line 1 (x1,y1) (x2,y2)
Line 2 (x3,y3) ...
3
votes
1answer
256 views
Why do definitions of distinct conic sections produce a single equation?
I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
3
votes
3answers
3k views
Orthogonal projection of a point onto a line
please give me a directions how to solve this:
find an orthogonal projection of a point T$(-4,5)$ onto a line $\frac{x}{3}+\frac{y}{-5}=1$
3
votes
3answers
659 views
How is the angle between 2 vectors in more than 3 dimensions defined?
I would like to know how the angle between two n-vectors is defined. I mean whether it is unique and how we may compute it (is the inner product a valid method in the n-dimensional space?). I have ...
2
votes
2answers
417 views
Find the area of a triangle using analytic geometry
Given are the points $P (1,0)$ and $Q (3,2)$. The points $P$ and $Q$ have the same distance to a certain line $l$, which intersects the positive x-axis in the point $A$ and the positive y-axis in the ...
2
votes
3answers
479 views
A good Open Source book on Analytic Geometry?
Hi my course specifically talks about :
Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, ...
2
votes
2answers
193 views
Conditions for intersection of parabolas?
What are the conditions for the existence of real solutions for the following equations:
$$\begin{align}
x^2&=a\cdot y+b\\
y^2&=c\cdot x+d\end{align}$$
where $a,b,c,d $ are real numbers.
...
1
vote
4answers
3k views
Horizontal tank with hemispherical ends depth to capacity calculation
I am trying to find an accurate way of calculating the capacity of an underground tank at a given depth. The tank manufacturer has provided a strapping table for the tank which tells me the capacity ...
0
votes
2answers
207 views
Problem with finding the equations of the lines tangent to a certain circle
This is a long question, and might seem like a repost of my earlier questions, but it isn't, hear me out:
In my book is written:
The equation of the line tangent to the circle $x^2+y^2=r^2$ in the ...
0
votes
2answers
147 views
Help me with Cylinder -coordinates problem, back to Cartesian or not? How to do it fast?
Source of the problem, 3b here.
Problem Question
Electricity density in cylinder coordinates is $\bar{J}=e^{-r^2}\bar{e}_z$. Current creates magnetic field of the
form ...
0
votes
2answers
88 views
Calculate Points for a Parallel Line
Given a line running through p1:(x1,y1) and p2:(x2,y2),
I need to calculate two points such that a new parallel line 20 pixels away from the given line runs through the two new points.
Edit: The ...
7
votes
4answers
994 views
What Does Homogenisation Of An Equation Actually Mean?
For example, if we have a conic;
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
What does homogenising this equation with another line (say ax + by + c = 0 ) actually mean? As in, what are the graphical ...
6
votes
3answers
3k views
Equation of angle bisector, given the equations of two lines in 2D
I have two lines in 2D expressed with general equation (or implicit equation):
First line: $a_1x+b_1y=c_1 \qquad(1)$
Second line: $a_2x+b_2y=c_2 \qquad(2)$
If the two lines are intersecting I will ...
5
votes
2answers
348 views
Find equation of quadratic when given tangents?
I know the equations of 4 lines which are tangents to a quadratic:
$y=2x-10$
$y=x-4$
$y=-x-4$
$y=-2x-10$
If I know that all of these equations are tangents, how do I find the equation of the ...
4
votes
1answer
183 views
Why it is sufficient to show $|f'(z)-1|<1$?
According to an article entitled "On the Univalency of Certain Analytic Functions" by Wang et al. (2006), we have to show that $|f'(z)-1|<1$ in order to find the radius of univalency for the class ...
4
votes
1answer
361 views
How to project the surface of a hypersphere into the full volume of a sphere?
The game I mentioned in "Navigating though the surface of a hypersphere in a computer game" is taking shape in here. The world is a 3-sphere where everything belongs. In Euclidean coordinates, for ...
4
votes
2answers
616 views
Find equation for hyperbola
Just taking (failing) a simple algebra class, can't figure this one out and no one can explain it to me and the book just tells me to do it.
Find an equation for the hyperbola described:
foci ...
4
votes
3answers
2k views
Find the coordinates in an isosceles triangle
Given:
A = (0,0)
B = (0,-10)
AB = AC
Using the angle between AB and AC, how are the coordinates at C calculated?
3
votes
1answer
28 views
Point-slope Equation
Let suppose that $R(x_1, y_1)$ is a point on the (x, y)plane and a line $L$ with slope $m$ passes through this point. There is a point $S(x_1, y_1)$ on $L$ such that $R$ and $S$ are coincident points. ...
3
votes
2answers
59 views
Difference of two points on a plane
If $P(x_1, y_1)$ and $Q(x_2, y_2)$ are the two points on a plane, then
the change in $x$ and $y$ coordinates is denoted by $∆x$ and $∆y$ respectively.
Therefore, $x = ∆x = x_2 - x_1$ and $y = ...
3
votes
4answers
1k views
How to calculate the two tangent points to a circle with radius R from two lines given by three points
I need to calculate the two tangent points of a circle with the radius $r$ and two lines given by three points $Q(x_0,y_0)$, $P(x_1,y_1)$ and $R(x_2,y_2)$.
Sketch would explain the problem more. I ...
3
votes
1answer
163 views
Recomputing arc center
Please excuse my poorly drawn doodle here, I'm almost inept at drawing.
I'm attempting to compute i2, j2, x2, y2.
Knowns:
x1, y1, xk, yk, i1, j1, the arc is circular
Constraints:
resulting arc ...
2
votes
3answers
80 views
Analytic geometry straight line problem
Prove that two straight lines represented by the equation $x^3+y^3+bx^2y+cxy^2=0$ will be at right angles if $b+c=-2$
I didn't know that even straight lines like planes can be represented by a ...
2
votes
1answer
96 views
Showing: point of polytope which maximizes the minimum distance to a vertex is a barycentre?
Let $T_1$ and $T_2$ be two regular $(n-1)$-dimensional simplices with vertices $$(t,0,\ldots,0), (0,t,\ldots, 0),\ldots, (0, 0, \ldots, t),$$ and $$(t-n+1,1,\ldots, 1), (1, t-n+1, \ldots, 1), \ldots, ...
2
votes
4answers
4k views
How to Determine an Equation of a Circle using a Line and Two Points on a Circle
My question goes like this:
Determine the equation of a circle tangent to the $x$-axis and passing through $(5,1)$ and $(12,8)$.
I need not only the answers, but also the steps on how you did it so ...
