Lecture 1. Ellipse
At this lecture, we will consider the concept of a second-order curve and study one of these curves in detail.
We have previously studied the first-order line which is called a straight line. It is given by an equation of the first degree. We define the concept of a second-order line.
A second-order line or curve is a planar figure that can be defined in a Cartesian coordinate system by a second-degree equation of this type (see the video), an equation from the variables x and y. In this case, it is required that among the coefficients a, b, c there is at least one that is not equal to zero, otherwise, if all these coefficients are reset to zero, we get an equation of the first degree.
Note that for some parameter values, this equation may not have a solution. For example, if the parameters a and f are 1 and the rest are 0, the equation will have no roots, i.e. in this case, we will get an empty set of points. The figure given by such an equation is conventionally called an imaginary one. In the future, we will consider the actual figures and classify them.
To do this, we will create two determinants shown in the video. The first one is denoted by Δ, the other – by δ.
If Δ ≠ 0, the figure is called non-degenerate. In this case, we can get an ellipse, in particular, a circle. Today, we will speak about this figure. Next, we may get a hyperbola. For example, a hyperbola is a graph of the inverse proportionality. And also a parabola, which is known to be the graph of a quadratic function.
If Δ = 0, the line is called degenerate. For example, an ellipse can degenerate into a point, which can result in a point defined, for example, by the equation shown in the video. Next, we can get two intersecting lines, such as these (see the video), or two parallel lines, which can coincide, in particular.
Today, we will take a closer look at a second-order line which is called an ellipse. Let’s give a general definition: an ellipse is a set of points in the plane, the sum of the distances from each of them to two fixed points is constant. These fixed points are called foci. Let the distance between the foci be 2c, this distance is called the inter-focus distance. Taking an arbitrary point M lying on the ellipse, we obtain by definition that the sum of the segments MF1 and MF2 is equal to some constant number, which we denote by 2a. In order for the ellipse to exist, i.e. the result is not an imaginary figure, but a real one, we require that a be greater than c. The segments MF1 and MF2 are called focal radii at the point M.
Let’s set several points that satisfy the specified equality. Each time the point changes, but the sum of the specified distances is constant (see the video). We get a certain figure, which is called an ellipse, while, as I’ve said, a > c, which follows from the triangle inequality. Now, let’s bring the two foci together. If we draw them closer, the figure will change a little (see the video) and in the limit, if we continue to bring them closer together, when the focus points merge, our ellipse will turn into a well-known curve – a circle (see the video). A circle is an ellipse whose two foci coincide. In this case, the radius of the circle is equal to the parameter a, and it coincides with the two focal radii of the ellipse.
Now, we consider an arbitrary ellipse and define a specially selected coordinate system, in which we derive the ellipse equation, i.e. we define the ellipse analytically. As the origin of coordinates, we consider the point O – the middle of the segment F1F2, the axis of the abscissa is directed along the points F1, F2, the Oy axis is drawn perpendicular to the axis of the abscissa since we have a Cartesian system. Note that the ellipse is symmetric with respect to the selected axes, so these axes are called the axes of symmetry of the ellipse, the point O is called the center of the ellipse. The considered coordinate system is called a conic system. In particular, if we have a circle, the axes can be chosen arbitrarily, and the origin is the center of the circle.
In the selected coordinate system, the foci get certain coordinates with abscissas –c and c, the ellipse itself intersects the abscissa axis at points a and –a. Now another parameter denoted by b is entered. In this case, the ellipse will intersect the ordinate axis at points -b and b. Parameter a is called the major semi-axis, parameter b is called the minor semi-axis. Note that b does not exceed the parameter a. These four points of intersection of the ellipse with the coordinate axes are called the vertices of the ellipse. If you draw straight lines parallel to the coordinate axes through the vertices, you will get a rectangle that completely encloses the ellipse. This rectangle is called a reference rectangle. The segments MF1 and MF2, which we called focal radii, can be calculated for any point M from the radius using the specified formulas (see the video).
Now, we derive the ellipse equation in the canonical coordinate system. We consider the ellipse. Let’s take an arbitrary point lying on the ellipse and make a characteristic equality according to the definition (see the video). Let’s write the distance from the point M to the foci, express them in terms of the coordinates of these points using the well-known formula, the distance between two points, and get this equality (see the video): there are two square roots on the left side of this equation. To get rid of them, you will have to square the equation twice, perform transformations, after which the equation will take the specified form (see the video). We remember that we denoted the difference a2–c2 by b2, this equality can be simplified by dividing it by the right part. We make the specified substitution and get the following equation (see the video). This equation is called the canonical ellipse equation.
Now again, let’s look at the ellipse, we already know by what equation it is given in the canonical coordinate system. Let’s take an arbitrary point on the ellipse, fix it, and draw an arbitrary secant through it. It will intersect the ellipse at some other point. We will bring this second point closer to the first one. In the limit, the second point will merge with the first one, i.e. we get the so-called limit position of the secant. This limit position of the secant is called the tangent to the ellipse at this point. The tangent equation can be set in the specified form (see the video). In particular, let’s consider a tangent at one of the vertices of the ellipse, for example, at a vertex with the coordinates (0, b). If we substitute the coordinates in the specified equation, we get the tangent, which is given by the equation y = b. In this way, we can show that the ellipse touches all sides of the reference rectangle.
Let’s consider an interesting optical property of an ellipse: if you put a radiation source in one of the foci, all the rays directed in different directions, when reflecting from the ellipse, will eventually gather in a different focus. Let’s look at the figure. Let’s take an arbitrary ray and reflect it (see the video). Let me remind you that the angle of reflection is equal to the angle of incidence. In this case, this ray will move to the second focus. Exactly the same will happen with all other rays. So, all of them will go to the second focus (see the video). Note that if you rotate an ellipse around an axis that passes through the foci, you will get a spatial figure, which is called an ellipsoid of rotation. For an ellipsoid, exactly the same property is true that we specified for a planar figure. Thus, we can say that the focus of an ellipse is a place where a large amount of energy is simultaneously concentrated.
Now, we introduce an important characteristic feature of an ellipse, it is called the eccentricity. This number is equal to the ratio of the parameters c and a, and since we have a > c, the eccentricity of the ellipse is always less than 1, but it is obviously a non-negative value.
Let’s look at some examples of ellipses and indicate what eccentricities they have. The slide shows three figures (see the video). You can see that if the eccentricity increases, i.e. the ratio of c to a will grow, the ratio of the parameters b to a will decrease, on the contrary, i.e. with increasing eccentricity, the ellipse will stretch along the abscissa axis. Please, note that the eccentricity on the left side of the figure is larger and the ellipse is more elongated than the one on the right side (see the video). Conversely, if the eccentricity decreases, the ellipse will look more and more like a circle, and in the limit when the eccentricity is reset to zero, we will get exactly a circle (see the video), i.e. an ellipse the foci of which coincide.
Let’s define one transformation called compression to a certain axis. Choose the abscissa axis as the axis and fix the compression ratio equal to the ratio of b to a. In this case, each point in the plane passes a new point. If we denote the original point with the letter M, we denote the new point with M’. let’s look at the figure: M is the original point with the coordinates (z, y), the new point has the coordinates (x’, y’). We define the coordinate correspondences as follows: the first coordinate does not change, and the second coordinate y’ increases, i.e. it changes k times, where k is the ratio b/a. From these equalities, we can express the value of the coordinates of the original point.
For example, let’s take a certain line, such as a circle, and translate each of its points to a new point using the specified transformation. In the figure, we can see that the points of the circle will cross at the point of some other line (see the video). We prove analytically that this line is an ellipse. To do this, in the canonical equation of a circle with radius a, we substitute the specified ratio instead of the variables, transform it, and, as we see, we get an equality that sets the equation of an ellipse with semi-axes a and b with respect to the variables x’, y’. Thus, any ellipse can be obtained using the compression from a circle.
Second: we know that a circle can be defined by a system of parametric equalities. If you replace the formulas shown on the slide with this system again, this system is converted to another system of equalities that will set the ellipse. Thus, an ellipse can also be defined by a parametrically specified system of equalities.