]> Functions and Geometry

### Chapter 1: Differentiation; Section 2: Real Functions; page 5

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# Functions and Geometry

## Implicit functions

Until now we have presented the function in the form  y = f(x) , which is called an explicit function. There are functions that cannot be expressed explicitly, but as implicit functions:

 $F\left(x,y\right)=0$ (1.2.5.1)

where  F(x,y)  is a mathematical expression of the variables  x  and  y .

As an example we can take the so called algebraic function:

 $F\left(x,y\right)={p}_{0}\left(x\right){y}^{n}+{p}_{1}\left(x\right){y}^{n-1}+....+{p}_{n-1}\left(x\right)y+{p}_{n}\left(x\right)=0$ (1.2.5.2)

where the expressions  ${p}_{k}\left(x\right)$  are polynomials of order k .

As we already know, if (1.2.5.2) includes powers $\ge 5$  of  x  and of  y , except in special cases; there is no way to solve it analytically, neither in terms of  x  nor in terms of  y .

On the other hand, any explicit function can be written in implicit form:

 $y=f\left(x\right)\text{ }\text{\hspace{0.28em}}\text{gives}\text{\hspace{0.28em}}\text{ }F\left(x,y\right)=y-f\left(x\right)=0$ (1.2.5.3)

From the definition of the inverse function, it follows that if the inverse of an implicit function is equal to the function itself, then  x  and  y  are interchangeable or

 $F\left(x,y\right)=F\left(y,x\right)$ (1.2.5.4)

As an example let's take the explicit double valued function

 $y=±\sqrt{\frac{5-{x}^{2}}{1+{x}^{2}}}$ (1.2.5.5)

whose domain is  [−√5,√5]. After taking the square of both sides and making a simple rearrangement, one obtains

 $F\left(x,y\right)={y}^{2}{x}^{2}+{y}^{2}+{x}^{2}-5=0$ (1.2.5.6)

which means that the inverse of function (1.2.5.5) is the function itself . However, if the function (1.2.5.5) was not double valued, e.g. restricted only to non-negative y values, then since the procedure includes squaring of some expressions; this restriction is lost, and (1.2.5.6) would not represent the restricted (1.2.5.5).

## Parametric presentation

There are functions that cannot be written explicitly, but by means of an additional parameter that connects between the variables (x and y) in the form:

 $\left\{\begin{array}{l}x={\text{f}}_{1}\left(p\right)\\ y={\text{f}}_{2}\left(p\right)\end{array}\right\}$ (1.2.5.7)

where  ${\text{f}}_{\text{1}}\left(p\right)$  and  ${\text{f}}_{\text{2}}\left(p\right)$  are explicit functions of the parameter  p .

Only if  ${\text{f}}_{\text{1}}\left(p\right)$  can be inverted, can one write the explicit functional relation:

 $y={\text{f}}_{2}\left({\text{invf}}_{1}x\right)$ (1.2.5.8)

Even if this is the case, there are some occasions when the parametric presentation could be more convenient to manage.

An example of the parametric presentation is the Cycloid:

 $\left\{\begin{array}{l}x=r\left(\alpha -\mathrm{sin}\alpha \right)\\ y=r\left(1-\mathrm{cos}\alpha \right)\end{array}\right\}$ (1.2.5.9)

where  α  is a parameter and  r  - a constant.

The cycloid is constructed and derived in Fig. Cycloid .

## Geometry and translation

Geometry deals with the shapes of objects. Some times we use geometrical properties in relation with functions. One has to understand the difference, in order to have a meaningful comparison between them.

The aspect of geometry that uses the coordinate system in order to position and deal with different objects is called Analytical Geometry. We are going to limit ourselves to a very elementary introduction of the geometry on a plane.

In order to present geometrical objects, both axes of the Cartesian coordinate system represent length, and the scales are identical. Different scales distort the shape of an object. The distance between two points $\left({x}_{1}\text{ },\text{ }{y}_{1}\right)$ and $\left({x}_{2}\text{ },\text{ }{y}_{2}\right)$ is given by

 $d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$ (1.2.5.10)

and the translation and/or rotation does not modify the properties of a geometrical object. The property of distance as defined by (1.2.5.10) is not limited only to geometrical space, but to the more general metric space. For example if we want to display a functional relation of energies, the notion of distance in this space could have (according to the circumstances) a meaning of energy difference. In case of a functional relation between physical entities with different physical dimensions, the space could not be metric.

Translation means moving all the points of an object by the same distance and in the same direction. If an object is translated by the distance of  Δx  in the  x  direction and by  Δy  in the  y  direction, then for any point of the object the coordinates  (x, y)  before the translation will become  (x', y')  related by

 $\left\{\begin{array}{l}x\text{'}=x+\Delta x\\ y\text{'}=y+\Delta y\end{array}\right\}$ (1.2.5.11)

A graphical example is given later on this page under the subtitle "Parabola".

The formal definition of rotation is given later on this page under the subtitle "Polar coordinates and rotation".

At this point it is important to stress the difference in dispalaying a function. The distortion of the shape of a function due the to use of different scales of the coordinate axes does not change the properties of a function. Translation or rotation change the properties of the function, since the dependence of the function on its argument is important, rather than the preservation of the geometrical shape.

The geometical objects we are going to discuss are

• Straight line
and the conic sections:
• Parabola
• Ellipse
• Hyperbola

## Straight line

A straight line corresponds to a linear relation between  x  and  y , which can be written in Cartesian coordinates as:

 $Ax+By+C=0$ (1.2.5.12)

where  A,B  and  C  are constants and can take any value, except  A=B=0 . The case of  A = 0  corresponds to a horizontal line,  B = 0  to a vertical line, and  C = 0  to a line passing through the origin.

The explicit presentation

 $y=kx+b$ (1.2.5.13)

where  k  and  b  are constants, does not include the vertical lines. By substituting  x = 0  in (1.2.5.13) one obtains that  b  is the  y-intercept  (x,y) = (0,b) .

The constant k can be expressed by the coordinates of any two different points belonging to the line according to the following procedure:

 $\begin{array}{l}{y}_{2}=k\text{ }{x}_{2}+b\\ {y}_{1}=k\text{ }{x}_{1}+b\end{array}\right\}\text{ }⇒\text{ }k=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}=\frac{\Delta y}{\Delta x}$ (1.2.5.14)

This makes k the slope of the line:

 $k=\mathrm{tan}\alpha$ (1.2.5.15)

where α is the angle between the line and the x axis. The Fig. Straight Line illustrates this point.

From this Fig. and by using (1.2.3.4) (tangent of an angle subtracted by another), one can express the tangent of the angle between two lines by:

 $\mathrm{tan}\beta =\frac{{k}_{2}-{k}_{1}}{1+{k}_{1}{k}_{2}}$ (1.2.5.16)

where  β  is the angle between the lines and the indexed  k's  are the slopes of the lines. The order of the lines in (1.2.5.16) is irrelevant, since the sign inversion of  tanβ  belongs to the other angle of intersection (complimentary to  π). One obtains from (1.2.5.16) that the condition for two lines to be perpendicular is:

 ${k}_{2}=-\frac{1}{{k}_{1}}$ (1.2.5.17)

It is elementary to verify that a line with slope k passing through a point  $\left({x}_{0}\text{\hspace{0.17em}},{y}_{0}\right)$  is:

 $y-{y}_{0}=k\left(x-{x}_{0}\right)$ (1.2.5.18)

From (1.2.5.14) and (1.2.5.18) one obtains that a line passing through two points  $\left({x}_{1}\text{ },{y}_{1}\right)$  and  $\left({x}_{2}\text{ },{y}_{2}\right)$  is

 $y-{y}_{1}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\left(x-{x}_{1}\right)$ (1.2.5.19)

We have thus far learned about the geometric properties of a straight line. In order to study the case of a linear function, we'll use the example of a motion with constant velocity, where the relation is

 $x=v\text{ }t+{x}_{0}$ (1.2.5.20)

where  t  is the time (the independent variable),  x  is the position along a straight line (the dependent variable),  v  is the constant velocity and  ${x}_{0}$  is the position at  t = 0 . In this example the variables have different physical dimensions and the notion of distance between two points (1.2.5.10) is meaningless.

The slope (1.2.5.14) becomes the constant velocity:

 $v=\frac{{x}_{2}-{x}_{1}}{{t}_{2}-{t}_{1}}=\frac{\Delta x}{\Delta t}$ (1.2.5.21)

Although graphically the velocity v is given by the ratio of two perpendicular segments  Δx  and  Δt , it does not really represent a tangent of an angle, but we are still going to call it the slope of the function independent of any physical dimensions. The angle of the line with the  t  axis depends on the scale of the coordinates, and therefore looses its meaning in case of a function. This is illustrated in Fig. Slope and angle.

## Parabola

Conic sections are obtained by intersecting a plane with a dual conic surface. Their general form is

 $A{x}^{2}+B{y}^{2}+Cxy+Dx+Ey+F=0$ (1.2.5.22)

where A, B, C, D, E and F are constants. Each one of the conic sections can be brought to a simple form called canonical, with the aid of appropriate translations and rotations. The conic sections are obtained in Appendix 02 using material of chapter 3. Its related illustration Fig. Conic sections can be oserved independently.

The canonical form of one of them - the parabola is

 ${y}^{2}=4ax$ (1.2.5.23)

where  a  is a constant. Any point of the parabola is at equal distance from the line  x = −a  (directrix) and from the point  (x,y) = (a,0)  (focus) . This property is illustrated in Fig. Parabola - canonical form, and its proof is given later in this page under the subtitle "Polar coordinates and rotation".

The canonical form (1.2.5.23) is a double valued function

 $y=\left\{\begin{array}{l}±2\sqrt{a}\sqrt{x}\text{ }\text{ }\text{for}\text{\hspace{0.28em}}a\ge 0\text{\hspace{0.28em}}\text{and}\text{\hspace{0.28em}}x\ge 0\\ ±2\sqrt{-a}\sqrt{-x}\text{\hspace{0.28em}}\text{\hspace{0.28em}}\text{\hspace{0.28em}}\text{for}\text{\hspace{0.28em}}a\le 0\text{\hspace{0.28em}}\text{and}\text{\hspace{0.28em}}x\le 0\end{array}\right\}$ (1.2.5.24)

The most common use of the parabola as a single valued function, is the second order (quadratic) polynomial

 $y={a}_{0}{x}^{2}+{a}_{1}x+{a}_{2}$ (1.2.5.25)

that can be obtained by an appropriate translation from  $y={a}_{0}{x}^{2}$  (see exercise 4), which is obviously a parabola. A graphic demonstration of the use of translations on a parabola is presented in Fig. Translation.

In the case of a linear motion with constant acceleration (1.2.2.7)

the parabolic function  (l)  and its argument  (t)  have different physical dimensions and the notions of distance (1.2.5.10) and angle are meaningless. A simple example is the motion of an object thrown horizontally under the influence of constant gravitation. The height  y  and the horizontal distance  x  as functions of time t are

 $\begin{array}{l}y={h}_{0}-\frac{g{t}^{2}}{2}\\ x={v}_{0}t\end{array}\right\}$ (1.2.5.26)
where:
• ${h}_{0}$ is the height at the beginning
• ${v}_{0}$ is the horizontal velocity
• g is the gravitational acceleration

From (1.2.5.26) after the exclusion of  t , one obtains the path of the object

 $y={h}_{0}-\frac{g}{2{v}_{0}^{2}}{x}^{2}$ (1.2.5.27)

Although (1.2.5.27) includes different physical dimensions, the function  y  and the argument  x  have the same dimension (length), and therefore the notion of distance (1.2.5.10) and of angle are meaningful.

## Polar coordinates and rotation

The polar coordinates are another way of defining a point in case of a two dimensional (planar) metric space. One of the coordinates is the distance from the origin (the radial distance), commonly denoted by  r  and expressed graphically by a straight segment connecting the origin with the point. The other one is the angle between the positive direction of the  x  axis with this segment. This angle is denoted usually by the small Greek letter $\phi$ or φ (phi), and is called the azimuth.

This same angle, with different name (α), was shown in Fig. Angles as a definition of an angle with respect to the cartesian coordinate system.

Any point on a plane can be expressed by a single valued pair of polar coordinates. Their domains are restricted for obvious reasons:

 $\begin{array}{l}r\ge 0\\ -\pi \le \phi \le \pi \end{array}\right\}$ (1.2.5.28)

The Cartesian coordinates, expressed by the Polar coordinates are:
 $\left\{\begin{array}{l}x=r\mathrm{cos}\phi \\ y=r\mathrm{sin}\phi \end{array}\right\}$ (1.2.5.29)
and the Polar - by the Cartesian:
 $\left\{\begin{array}{l}r=\sqrt{{x}^{2}+{y}^{2}}\\ \mathrm{cos}\phi =\frac{x}{r}\\ \mathrm{sin}\phi =\frac{y}{r}\end{array}\right\}$ (1.2.5.30)

At the origin of the Cartesian coordinates,  r = 0 , and $\phi$ is undefined. This of no significance, since for this unique point the value of the azimuth is irrelevant. The relations between the Cartesian and the polar coordinates are shown graphycally in Fig. Relations between coordinates.

A parabola (1.2.5.23) with  a > 0  expressed by polar coordinates with the focus situated at the origin is

 $r=\frac{2a}{1-\mathrm{cos}\phi }$ (1.2.5.31)

The user should carefully follow Fig. Polar coordinates, where (1.2.5.31) is obtained and the focal properties of the parabola are proven.

Polar coordinates provide a simple representation of the rotation (in two dimensions) about the origin. In order to rotate an object by an angle  Δφ , any point with an angular coordinate  φ  will move to the angle  φ'  by the following relation:

 $\phi \text{'}=\phi +\Delta \phi$ (1.2.5.32)

A simple example is a straight line inclined by an angle   ${\phi }_{0}$   toward the  x  axis and starting from the origin. Its presentation in polar coordinates is   $\phi ={\phi }_{0}$  . The substitution of  φ  from (1.2.5.32) yields the presentation of the new line:   $\phi \text{'}={\phi }_{0}+\Delta \phi$  .

By using equation (1.2.5.32) one can also obtain the relation between the Cartesian coordinates of the object before and after the rotation. From (1.2.5.29) and considering that the rotation does not modify the polar coordinate  r , one can write that after the rotation the Cartesian coordinates of the object will be:

 $\left\{\begin{array}{l}x\text{'}=r\mathrm{cos}\phi \text{'}\\ y\text{'}=r\mathrm{sin}\phi \text{'}\end{array}\right\}$ (1.2.5.33)

By substituting  φ'  from (1.2.5.32) into (1.2.5.33) and by using (1.2.5.29), one obtains

 $\left\{\begin{array}{l}x\text{'}=r\left(\mathrm{cos}\phi \mathrm{cos}\Delta \phi -\mathrm{sin}\phi \mathrm{sin}\Delta \phi \right)=x\mathrm{cos}\Delta \phi -y\mathrm{sin}\Delta \phi \\ y\text{'}=r\left(\mathrm{cos}\phi \mathrm{sin}\Delta \phi +\mathrm{sin}\phi \mathrm{cos}\Delta \phi \right)=x\mathrm{sin}\Delta \phi +y\mathrm{cos}\Delta \phi \end{array}\right\}$ (1.2.5.34)

In a similar way one obtains from the same equations the reversed relation

 $\left\{\begin{array}{l}x=y\text{'}\mathrm{sin}\Delta \phi +x\text{'}\mathrm{cos}\Delta \phi \\ y=y\text{'}\mathrm{cos}\Delta \phi -x\text{'}\mathrm{sin}\Delta \phi \end{array}\right\}$ (1.2.5.35)

An example of rotation using the Cartesian coordinates is presented laterer on this page under the subtitle "Hyperbola".

## Ellipse

The ellipse is another conic section and its canonical form is

 $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\text{ }\text{with}\text{ }a\ge b>0$ (1.2.5.36)

or in parametric form

 $\left\{\begin{array}{l}x=a\mathrm{cos}\theta \\ y=b\mathrm{sin}\theta \end{array}\right\}\text{\hspace{0.28em}}\text{where}\text{\hspace{0.28em}}\text{\hspace{0.28em}}a\ge b>0$ (1.2.5.37)

The constants  a  and  b  are called the big and the small radii correspondingly. In the case of the equality  a = b , the ellipse becomes a circle. The proof of the equivalence of (1.2.5.36) and (1.2.5.37) are left as a simple exercise for the user.

The ellipse in its canonical form is constructed from its parametric presentation in Fig. Ellipse - canonical form .

The ellipse has two focal points  (f,0)  and  (−f,0), where

 $f=\sqrt{{a}^{2}-{b}^{2}}$ (1.2.5.38)

The sum of the distances of any point from the ellipse to the foci is constant, and equals  2a . The proof of this property is a partial requirement of the exercise 5 (see below).

These focal properties are exhibited and used for a simple construction of the ellipse in Fig. Ellipse - focal properties .

Here is another example using the polar coordinates. In some cases it is useful to express a circle whose circumference passes through the coordinate's origin in polar coordinates. We want to translate the circle  ${x}^{2}+{y}^{2}={a}^{2}$  upwards on the  y  axis by  $\Delta y=y\text{'}-y=a$ , which brings the circle to the form

The substitution of  x' = rcosφ  and  y' = rsinφ  yields the final result

 $r=2a\mathrm{sin}\phi \text{ }\text{with}\text{ }\text{0}\le \phi \le \text{π}$ (1.2.5.39)

The limitation on  φ  is necessary in order to keep  r  non-negative. Actually the points corresponding to  φ = 0  and to  φ = π  represent the same point.

## Hyperbola

The last conic section is the hyperbola with the canonical form of

 $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\text{ }\text{with}\text{ }a\ne 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b\ne 0$ (1.2.5.40)

The hyperbola is double valued (except at $x=±|a|$) and the domain consists of two intervals: $x\ge |a|\text{ }\text{and}\text{ }x\le -|a|$ . The points where  y = ±b  correspond to  x = ±a2 , as can be seen in (1.2.5.40).

The hyperbola also has a parametric form:

 $\left\{\begin{array}{l}x=a\mathrm{cosh}p\\ y=b\mathrm{sinh}p\end{array}\right\}\text{ }\text{with}\text{ }a\ne 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b\ne 0$ (1.2.5.41)

which exhibits again the similarity between the trigonometric and the hyperbolic functions. From the parametric form (1.2.5.41) one easily obtains the presentation (1.2.5.40), however, since this procedure includes taking a square, the expressions are not entirely equivalent. Since the cosh by definition is positive, the parametric presentation includes only the interval of  x , which sign corresponds to that of  a .

The hyperbola has two focal points  (f,0)  and  (−f,0), where

 $f=\sqrt{{a}^{2}+{b}^{2}}$ (1.2.5.42)

The absolute value of the difference between the distances of any point from the hyperbola to the foci is constant and equals  2|a|. The proof of this is very similar to that of the focal property of an ellipse, which is given here as an exercise.

Graphical display of the hyperbola with emphasis on some of its properties are given in Fig. Hyperbola . It also contains a graphical presentation of the rotation of a hyperbola that is discussed in the following example.

An example of rotation.

The function

 $y=\frac{k}{x}$ (1.2.5.43)

where k is a constant is single valued. We'll prove that (1.2.5.43) is a rotated hyperbola with the following canonical form

 ${x}^{2}-{y}^{2}=2|k|$ (1.2.5.44)

meaning that  ${a}^{2}={b}^{2}=2|k|$ .

In the case of  k>0  we'll rotate (1.2.5.43) by   $\Delta \phi =-\frac{\pi }{4}$   and therefore

The substitution in the formulae of rotation (1.2.5.35) yields:

which substituted in (1.2.5.43) yields finally:   $x{\text{'}}^{2}-y{\text{'}}^{2}=2k$  , according to (1.2.5.44).

The case of  k<0  requires rotation of   $\Delta \phi =\frac{\pi }{4}$  , and the proof remains as a simple exercise for the user.

## Exercises

Exercise 1. Follow the procedure of Fig. Cycloid (1.2.5.9) and derive the parametric presentation of a point situated at a distance R from the center of the wheel (r remains the radius of the wheeel).

• R=r
• R=0
• r=0

Exercise 2. Given a straight line in its general form (1.2.5.12) and a point $\left({x}_{0},{y}_{0}\right)\text{\hspace{0.28em}}$ outside of the line, calculate the shortest distance between the point and the line! Hint: (1.2.5.17)

Exercise 3. The centre of a circle with radius r is at the origin of the coordinate system. A straight line is given in its general form (1.2.5.12).

1. What is the condition for the line not to have any common points with the circle, and what are the closest points of the line and of the circle in this case?
2. What is the condition for the line to have only one common point with the circle, and what is this point?
3. What is the condition for the line to intersect the circle, and what is the length of the shortest arc obtained?

Note: This exercise is simpler if some of the results from the exercise 2 are used.

Exercise 4. For the parabolic function (1.2.5.25)

1. What is the necessary translation (Δx and Δy) for expressing it as
2. If the physical dimensions of  x and y  are  [x] and [y]  respectively, what are the dimensions of the constants  ${a}_{0},\text{\hspace{0.28em}}{a}_{1}\text{\hspace{0.28em}}\text{and}\text{\hspace{0.28em}}{a}_{2}$ ?
3. What are the dimensions of Δx, Δy and  $\sqrt{{\left(\Delta x\right)}^{2}+{\left(\Delta y\right)}^{2}}$ ?

Exercise 5. Prove the focal properties, as they are stated on this page in the following cases, by using the Cartesian coordinates:

1. Parabola. See (1.2.5.23) with the sentence that follows.
2. Ellipse. See (1.2.5.38) with the sentence that follows.

Exercise 6. The following expression in polar coordinates

represents the path of a planet around the sun, obtained from Newtonian mechanics. Show that this is an ellipse with one of the foci at the coordinate's origin (the sun) in the following way:

1. Translate one of the foci of the ellipse given in its canonical form (1.2.5.36) to the origin!
2. Express this shifted ellipse in polar coordinates!
3. What are the values of β and A in terms of the radii a and b?
4. What is the meaning of the sign of β ?

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