Questions tagged [reflection]
Reflection is a transformation that fixes a line or plane or a more general subset. Reflections appear in geometry, linear algebra, complex analysis, differential equations, etc -- therefore, this tag must be used with a tag describing the area of mathematics.
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Get normal vector as a function of vector and its reflection off said plane
This question was asked almost nine years ago, and I found it because I'm making a robot that will direct a mirror to reflect light at a desired angle. I need a little more information than what the ...
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Proving Conjugacy of Isomorphic Finite Reflection Groups via an Isometry in Orthogonal Group $ O(V) $
Let $ W_1 \text{ and } W_2 $ be finite reflection group acting on the euclidean space $ V $. If there exists an isomorphism between $ W_1 \text{ and } W_2 $ induced by the isometry $\varphi $ of $ V $...
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How to "reflect and combine" (mirror?) a function about an arbitrary plane?
I am looking for a general method to "reflect and combine" a function about an arbitrary plane defined by a normal vector through the origin, in both 2D and 3D. Ideally, the method should be ...
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Isogonal lines and an ellipse
Take two lines $l_1$ and $l_2$ with intersection $A$.
We say the lines $AM$ and $AN$ are isogonal with respect to the lines $l_1$ and $l_2$ if $AM$ is the reflection of $AN$ about the bisector of an ...
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isometries as products of reflections [closed]
I am taking a course on geometric transformations in the 2-dimensional Euclidean plane. I have been told that all isometries (length-preserving transformations) are products of reflections about ...
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Problem with understanding the proof using a reflection in the plane
I'm trying to understand the proof of following statement: The point M belongs to the diameter $\overline{AB}$ of the given circle. The chord $\overline{CD}$ passes through the point M and forms an ...
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Where does the term "reflection" come from?
Earlier today, I was asked why a motion of the plane that fixes a line of points is called a reflection and I was stumped for an answer.
The best explanation I can think of is that the image of a ...
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Jacobian of the vector reflection operator
While re-deriving some equations relevant to Monte-Carlo path tracing (specifically, the probability distribution of sampling a specific light direction from Sampling the GGX Distribution of Visible ...
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Confused on the result of Sequence of Geometric Transformations
Here is the question. Consider the parent function $f(x)=\frac{1}{x}$. Now do the following sequence of transformations.
$1.$ Shift up by $4$ units.
$2.$ Shift left by $2$ units.
$3.$ Vertically ...
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Show f an isometry with a fixed point in $\mathbb{R}^2$ is a composition of 1 or 2 reflections
Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be an isometry with a fixed point $A \in \mathbb{R}^2$. That is, $f(A) = A$. Show that $f$ is a composition of $1$ or $2$ reflections.
Here is my attempt. I ...
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Reflecting point across bisector of acute angle leads to perpendicular elsewhere
Exercise. Let $\angle BAC$ be an acute angle. In the interior of $\angle BAC$ there is a point $P$ whose projections onto the sides $AB$ and $AC$ are precicely $B$ and $C$.
Draw the bisector of the ...
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How to compute $RA$ in $O(n^2)$ operations instead of $O(n^3)$ using Householder reflection
I am currently writing a program that performs QR decomposition on a matrix $A$.
The guidelines to my assignment tell me that once I calculate $R$ using Householder reflections, there is a way to ...
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Describing the Eigenspace of a Linear Transformation on $\Bbb R^3$ that Rotates Points About a Line Through the Origin
Let A be the matrix of the linear transformation $T$. Without writing A, find an eigenvalue of A and describe the eigenspace in the following situations.
a. $T$ is the transformation on $\Bbb R^2$ ...
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Find the signature of a reflection.
I am doing some self-study and I have this task:
Show that the signature of the following mapping, the reflection about the hyperplane $a^{\perp}$, given by
$S_a(v) = v - 2\frac{<v,a>}{<a,a&...
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Bath towel on the spheric rope: minimize the area of self-intersection of a 'folded' spheric rectangle
Some time ago I was curious about a question related to my bath towel, which I hang on a rope to have fun (you can use your own towel to do this experiment in bath-o if you want):
'There is this ...
