A.1 Affine Coordinates.¶
- Affine Geometry
A geometry in which the Euclidean parallel postulate holds.
- Slope of line \(l\):
The unique \(m\in\mathbb{R}\) such that, for any two points \(P(x_1,y_1)\) and \(P(x_2,y_2)\) on \(l\),
Affine coordinates can be created for any (non-finite) affine geometry, and they will have the properties we associate with Cartesian Coordinates. The SMSG Axioms provide everything needed to create the typical Cartesian Coordinate System which can, with a bit more work, include the \(z\)-axis as well.
Distance in a Coordinate System \(O_{xy}\)¶
At the moment, distances have only been defined along the axes. Notice that the value of the coordinates of point \(P(x,y)\) are determined by creating a rectangle and measuring distances on the \(x\)- and \(y\)-axis using the SMSG Ruler Axioms. The next two theorems will establish the validity of the Euclidean distance metric.
Theorems¶
For two distinct points with the same \(y\)-coordinate, say, \(P(x_1,y)\) and \(Q(x_2,y)\), the distance between \(P\) and \(Q\) is
\[\text{Dist}(P,Q)=|x_1-x_2|\]For two distinct points with the same \(x\)-coordinate, say, \(P(x,y_1)\) and \(Q(x,y_2)\), the distance between \(P\) and \(Q\) is
\[\text{Dist}(P,Q)=|y_1-y_2|\]Note
From the algebraic properties of real numbers, we know that for any distinct \(a,b\in\mathbb{R}\)
\[|b-a|^2=|a-b|^2=(a-b)^2\]which makes the distance formula much easier to establish.
Distance Formula. Prove that, for any two distinct points \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) in the coordinate plane,
\[\text{Dist}(P,Q)=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\]Hint
The distance formula is simply the anlaytic geometry version of the Pythagorean Theorem.
Midpoint Formula. Prove that the midpoint \(M\) of line segment \(\overline{AB}\) where \(A(x_1,y_1)\) and \(B(x_2,y_2)\) is given by:
\[M(x,y)=\left( \frac{x_1+x_2}{2},\frac{y_1+y_2}{2} \right)\]Linear Equations. For any line \(l\) not parallel to the \(y\)-axis, prove all points \(P(x,y)\) satisfy the relationship \(y=mx+b\) for some \(m,b\in\mathbb{R}\).
Note
Lines parallel to the \(y\)-axis have equations of the form \(x=k\) for some \(k\in\mathbb{R}\). We say the slope of such lines is undefined. Some textbooks refer to these types of lines as having “no slope.” This is very confusing especially for high school students: “no slope” seems almost the same as “zero slope” due to how we communicate in real life. In this course, we will always refer to these lines as having undefined slope.