All Questions
35
questions
1
vote
0
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40
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Is there an isotomic analogous of circular points of infinity?
In isogonal pivotal (with pivot at the line of infinity) cubics with respect to a triangle $\triangle ABC$. By a suitable projective transformation, fixing $A$,$B$, and $C$, sending the incenter to ...
- 37
1
vote
0
answers
59
views
Calculate on which side of a cuboid is a given point located?
Correct me if I'm using incorrect terms, I'm not well-versed with geometrical terminology
I'm writing a code in which
I have a point &
I want to identify if the points lies on front, back, top, ...
- 111
5
votes
0
answers
115
views
There is a compass-like tool that can draw $y=x^2$ on paper. Is there one for $y=x^3$?
Is there a tool that can draw $y=x^3$ on paper?
I'm referring to low-tech tools, e.g. not computers.
I only know of tools that can draw $y=x^2$. The YouTube video "Conic Sections Compass" ...
- 25.7k
6
votes
2
answers
176
views
Why do equilateral triangles relate to cubics
I found this question talking about the relation between an equilateral triangle and cubics with three distinct real roots.
Here's an image from the original post with an example:
What this post says ...
1
vote
0
answers
79
views
Asymptotes of general cubic
I'm trying to understand why the following method seems to work in finding the asymptotes of a general cubic.
Take this cubic equation:
$$x^3+2xy^2-\frac{1}{4}y^3+6x^2-4xy+2y^2+7x-2y+1=0$$
This looks ...
- 36
2
votes
0
answers
74
views
Why this cubic surface just can find 3 lines?
I often heard that each smooth cubic surface contains even $27$ straight lines, such as this statement. So I use the Mathematica plot the equation with this code:$$\begin{align}1 + 54 x y z - &9 (...
- 604
0
votes
1
answer
122
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Further information on the reduction of cubic equations to a system of two conic sections
This question follows on from one I have previously asked, How to separate cubic equations into two conic sections: Deep dive into Omar Khayyam and I now would like some further advice on some ...
- 55
0
votes
1
answer
110
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Trajectory of a 2D constant jerk motion
This thread is a natural prosecution of this other.
Problem
Let $\xi,\eta$ be the horizontal and vertical coordinates of a plane. Consider the following sequence of point
\begin{equation}\begin{...
- 729
1
vote
1
answer
302
views
Solving cubic equations with sine and cosine sums.
I was playing with math, and then I tried to rewrite some cubic equation with sine power reduction formula
$$y^3 + my^2 + ny + d = 0.$$
Let
$$y = \sin(x).$$
Then
$$y^2 = \frac{1 - \cos(2x)}{2},$$
$$y^...
3
votes
0
answers
380
views
Solving a cubic equation using a rotated and scaled equilateral triangle
There is this property of a cubic equation with 3 distinct real solutions where you can draw an equilateral triangle above the graph (center lined up with the inflection point), and it can always be ...
- 65
1
vote
2
answers
146
views
When are the roots of a polynomial of degree 3 aligned?
Let $P \in \mathbb{C}[X]$ be a polynomial of degree 3. On what condition on the coefficients of $P$ are the three roots of $P$ aligned ?
To make things easier, we may assume that $P$ can be written as ...
- 459
1
vote
1
answer
209
views
Checking if 8 points in the projective plane lie on a singular cubic.
I need to check if 8 points of $\mathbb{P}^2$ (over a finite field) lie on a singular cubic with one of them a double point.
I know that to check if a point is singular we suffice to compute the ...
- 187
0
votes
1
answer
157
views
Determine the closest point(s) from a line to a parametric cubic curve where the distance is less than $L$
Given a parametric cubic curve in 2D space:
$$\vec r(t) = \vec At^3 + \vec Bt^2 + \vec Ct + \vec D$$
and a line defined by the equation:
$$px + qy + k = 0$$
I can easily determine where these two ...
- 418
1
vote
1
answer
129
views
Deducing a cissoid equation using an intersecting linear equation (Stillwell, 2001)
Exercise 2.5.2 in Mathematics and it's History 2nd ed. by John Stillwell asks to reader to deduce the equation for a cissoid using the linear equation;
$$Y=\frac{\sqrt{1-x^2}}{1+x}(X-1)$$
The line, ...
0
votes
2
answers
115
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Is this the Tucker-Gergonne-Nagel cubic ? or some other one
$M$ is the intersection of 3 cevians in the triangle $ABC$.
$$
AB_1=x,~CA_1=y,~BC_1=z.
$$
it can be easily proven that for both Nagel point and Gergonne point the following equation is true:
$$
S=\...
- 199