Talk:Line (geometry)/Archive 3

NOTE: As expected, one person is against changing anything. This draft is similar to other geometry articles posted on Wikipedia, as most readers of Wikipedia know. This article is on GEOMETRY. The text herein is about the geometry of the normal form.

The normal form of a line in ${\displaystyle \mathbb {R} ^{2}}$ is specified as the set

${\displaystyle L=\{(x,y):\mu x+\nu y=\rho \}}$,

where ${\displaystyle \mu ,\nu ,}$ and ${\displaystyle \rho }$ are constants such that ${\displaystyle \mu ^{2}+\nu ^{2}=1.}$

Geometrically, ${\displaystyle L}$ is a line tangent to a circle centered at the origin (denoted as the point ${\displaystyle O}$) with a given radius ${\displaystyle |\rho |}$. Consequently, this circle is the set ${\displaystyle C=\{(x,y):x^{2}+y^{2}=\rho ^{2}\}}$.

• The point of tangency of ${\displaystyle L}$ on ${\displaystyle C}$ is often called the foot of ${\displaystyle L}$ and is denoted as ${\displaystyle R}$. ${\displaystyle R(r_{x},r_{y})}$ has the coordinates ${\displaystyle r_{x}=\rho \mu }$ and ${\displaystyle r_{y}=\rho \nu }$.
  • ${\displaystyle R}$ is on ${\displaystyle C}$ because ${\displaystyle r_{x}^{2}+r_{y}^{2}=\rho ^{2}.}$
  • ${\displaystyle R}$ is on ${\displaystyle L}$ because ${\displaystyle \mu r_{x}+\nu r_{y}=\rho \mu ^{2}+\rho \nu ^{2}=\rho (\mu ^{2}+\nu ^{2})=\rho }$.

• The distance ${\displaystyle OR}$ is the length of a radius of ${\displaystyle C}$, so ${\displaystyle OR}$=${\displaystyle |\rho |}$. ${\displaystyle |\rho |}$ is often called the pedal distance of ${\displaystyle L}$, and by definition, is the distance of ${\displaystyle L}$ is from the orgin ${\displaystyle O}$.

• ${\displaystyle L}$ can specified in normal form as

${\displaystyle L=\{(x,y):{\frac {r_{x}}{\rho }}x+{\frac {r_{y}}{\rho }}y=\rho \}}$ or in standard form as ${\displaystyle L_{s}=\{(x,y):r_{x}x+r_{y}y=\rho ^{2}\}}$.^[1]

• The line containing the radius of ${\displaystyle C}$ at ${\displaystyle R}$ is perpendicular (in other words, orthogonal) to ${\displaystyle L}$, and is denoted as ${\displaystyle N_{O}}$, where

${\displaystyle N_{O}=\{(x,y):y={\frac {r_{y}}{r_{x}}}x\}}$ or ${\displaystyle N_{O}=\{(x,y):-r_{y}x+r_{x}y=0\}}$.

${\displaystyle N_{O}}$ is said to be the normal line of ${\displaystyle L}$ through the origin.

Since ${\displaystyle \{(\mu ,\nu ):\mu ^{2}+\nu ^{2}=1\}}$ specifies a unit circle with center at the origin, the coefficients may be specified as ${\displaystyle \mu =\cos \varphi }$ and ${\displaystyle \nu =\sin \varphi }$, that allows the normal form to be specified as

${\displaystyle L=\{(x,y):x\cos \varphi +y\sin \varphi =\rho \}}$,

where ${\displaystyle \tan \varphi ={\frac {\nu }{\mu }}={\frac {\sin \varphi }{\cos \varphi }}}$ and ${\displaystyle \varphi =atan2(v,u)=atan2(\sin \varphi ,\cos \varphi )}$.

Note atan2 is the angle the point ${\displaystyle (u,v)}$ makes counter-clockwise with the x-axis in the range ${\displaystyle -\pi \leq \varphi <\pi }$.

• ${\displaystyle \varphi }$ is often called the pedal angle of ${\displaystyle L}$, where the foot's coordinates are ${\displaystyle r_{x}=\rho \cos \varphi }$ and ${\displaystyle r_{y}=\rho \sin \varphi }$.

Since the normal form is a special case of the standard form, the geometric relationships of a line inherent in its normal form can be revealed for any line by a process called normalization. A line specified as ${\displaystyle \{(x,y):Dx+Ey=F\}}$ can be specified in normal form as ${\displaystyle \{(x,y):\mu x+\nu y=\rho \}}$ by dividing both sides of the equation by ${\displaystyle {\sqrt {D^{2}+E^{2}}}}$, such that

${\displaystyle \mu ={\frac {D}{\sqrt {D^{2}+E^{2}}}};\nu ={\frac {E}{\sqrt {D^{2}+E^{2}}}};}$ and ${\displaystyle \rho ={\frac {F}{\sqrt {D^{2}+E^{2}}}}}$.^[2]

Geometric relationships revealed by the normal form are identical to the relationship of the line with the polar axis in polar coordinates. It is not surprising that the normal form is easily coverted to a polar equation through the standard set of transform equations.^[3] 69.138.197.204 (talk) 23:05, 2 March 2021 (UTC)

As said in my second edit summary, the beginning of this text is blatantly wrong: the normal form is not a set. This could be easily fixed, but this text has multiple other issues. This section is about a specific equation of a line in an article devoted to the introduction of the concept of a line. So the use of more advanced concepts of geometry such as "radius of a circle", "tangent line" and "perpendicular" are logically incoherent, as their definition requires the concept of a line; this makes the proposed text confusing for some readers. There are other issues in this text, but it is not worth to enumerate them, since the two above issues are sufficient for asserting that this version of the section does not improve the current version, and is therefore not acceptable for Wikipedia. D.Lazard (talk) 10:28, 3 March 2021 (UTC)

Please clarify. I believe the level of reader knowledge of GEOMETRY is being kept far too low, especially in light of Wikipedia's vast array of GEOMETRY articles. Besides, ALL LINES ARE SETS OF POINTS -- yes the normal to a line is a line and, consequently, A SET OF POINTS with its own SPECIFICATION. In addition, why are the terms "radius of a circle" and "perdendicular" confusing in a GEOMETRY article? (How can you not mention "perpendicular" on a article on "normal" form?) The current article lacks information to understand the normal form. Readers who look at the article are in high school math and understand those terms. (I was going to prepare a diagram -- none of the sections contains them and the article needs them.) If you open your own reference, there is a diagram that accompany's the text. The vocabulary I mentioned above is still necessary for readers to understand the geometry of the normal form. OK -- apply your criteria to the main Hesse normal form article? It would be devastating if the criteria you use here are applied to it. And what would you do to article on the Real Line?

The Wikipedia editors regard this article as START class. It clearly needs continued improvement. The editors that know mathematics know what they expect out of an article on the geometry of lines. I do give compliments to the changes made on the line in polar coordinates section. However, the narrative still avoids terms readers are encountering in textbooks and their school classes. Looking at the talk and age of talk on this page, it appears many, many editors have given up trying to help. (Editors, please don't stop!) 2601:140:8980:4B20:AC68:BB15:9BFE:C2DD (talk) 16:23, 3 March 2021 (UTC)