Understanding Ellipses: Major vs. Minor Axis
Comprehensive Definition, Description, Examples & RulesÂ
Introduction
Ellipses are special shapes in geometry that look like elongated versions of circles. A unique property of ellipses is that when you add up the sum of distances from an end on the ellipse to the foci or two stable ends, it will be constant. Ellipses have a crucial part to play in the field of geometry, like major axes, minor axes, foci, and more. These geometrical functions make ellipses useful in an array of everyday applications in fields like astronomy, optics, architecture, engineering, etc.Â
What is an Ellipse?
An ellipse is the midpoint of all the ends on a plane whose distances from two stable ends in the plane are stable. The stable ends, which are encompassed by the curve, are known as foci (singular focus). The constant ratio is the eccentricity of the ellipse and the stable line is the directrix. Eccentricity is a concept of an ellipse that shows elongation and is symbolized by the letter ‘e’.
In other words, In the context of locus, an ellipse is a collection of all ends on an XY-plane whose distance from two stable ends (called foci) sums up to a constant value. An ellipse is formed when a cone is cut at an angle with its base. The conic section we get as a result is an ellipse.Â
Key Elements of an Ellipse
Ellipses have two axes along the x and y axes:Â
- Major Axis
- Minor Axis
The major axis is the ellipse’s longest diameter (represented by ‘a’), which runs through the midpoint from one end to the other, at the broadest section of the ellipse. On the other hand, the minor axis is the ellipse’s shortest diameter (denoted by ‘b’), passing through the midpoint at its shortest end. Half of the major axis is the semi-major axis, while half of the minor axis is the semi-minor axis.
Major Axis in an Ellipse
The longest straight line that passes through the midpoint of the ellipse is the major axis of the ellipse. It connects the two ends that are at the farthest distance from each other on the ellipse’s boundary. The major axis of an ellipse is responsible for defining its shape, size, and orientation, as well as attributing various unique properties to the ellipse along the way. An ellipse’s eccentricity or the measure of deviation from a perfect circle is also determined by the major axis. Moreover, the major axis also comes in handy in determining conic sections which, in turn, plays a huge role in important fields like astronomy, engineering, physics, and the like.Â
Minor Axis in an Ellipse
The minor axis of an ellipse is the shortest diameter that passes through the midpoint of the ellipse and is at right angles, or perpendicular to the major axis. It works along with the major axis to determine the unique shape of the ellipse, as well as giving an ellipse its distinguishing proportions and dimensions. In simpler words, the length of the minor axis gives an ellipse a ‘stretched’ or ‘squished’ shape when compared to a perfect circle. Mathematically we can say that the minor axis of an ellipse influences the eccentricity of an ellipse, which is a measure of deviation from the perfect circle or circularity.
Semi-Major Axis of an Ellipse
As the name suggests, the semi major axis ellipse is half of its major axis in length. It begins from the midpoint of an ellipse and ends at the farthest end on the ellipse’s boundary. A semi-major axis is one of the most fundamental concepts related to ellipses which gives ellipses their unique features and properties like size, orientation, etc. it also comes in handy in solving problems we encounter in everyday applications like astronomy, celestial mechanics, engineering, physics, etc.
Formula for Semi-Major Axis
The semi-major axis of an ellipse is the distance between the ellipse’s midpoint and its farthest edge end. It can also be defined as half of the length of the major axis. The following formula is used to calculate the length of the semi-major axis:Â
Semi-Major Axis = Half of the length of the Major Axis
Let’s assume that B and C are foci or the two stable ends encompassed by the curve. So, the semi major axis formula with A as any random end on the ellipse is:
Semi Major Axis = (AB +AC)/2
Or, in simpler words, Semi-Major Axis = a + b / 2 ( a and b are major and minor axes respectively).Â
Differences Between Ellipse and Oval
Ellipses and ovals are easily confused and mistaken terms. However, it is important to note that even though these shapes may appear similar, they both have unique properties that make them very different from each other. Go through the following important differences in ellipse vs oval to avoid confusion between the two:
- Ellipses are conic sections with eccentricities (e) ranging from 0 to 1, but ovals are geometrical forms that are not clearly described in mathematics.
- An oval is always an ellipse, but an ellipse is not necessarily an oval.Â
- An ellipse will always have two symmetric axes (semi-major axis and semi-minor axis). On the other hand, ovals can also have only one or two.Â
The Ellipse Function
The ellipse function can be used to express an ellipse in terms of algebra. We can use the following formula to compute the equation of the ellipse:
The equation for the midpoint of the ellipse at (0,0): x^2/a^2 + y^2/b^2 = 1
The equation for the midpoint of the ellipse at (h,k): (x-h)^2/a^2 + (y+k)^2/b^2 = 1
The concept of ellipse function finds use in an array of everyday applications to make our lives easier. Artists use the equation to create intricate designs. It is also applied in statistics and data representation to symbolize huge amounts of data pictorially. Ellipses are also seen in architectural designs that we come across daily like arches, domes, etc.Â
Understanding Ellipse Duality: Ellipsed
The concept of ‘ellipsed’ or ellipse duality is a principle in geometry that relates an ellipse to its polar ellipse. When the major axis of an ellipse aligns with the polar line of the polar ellipse, and both the ellipses generate from a focus, we can refer to the situation as an ellipsed or dual ellipse. At the same time, the minor axis of the ellipse aligns with the axis of the polar ellipse. This property of duality allows for reciprocal or inverse relationships and properties between ends and lines on the two shapes. This property also has a crucial role to play in fields like architecture, astronomy, engineering, and more.
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Key Takeaways
- An ellipse is the midpoint of all the ends on a plane whose distances from two stable ends in the plane are stable. The stable ends, which are encompassed by the curve, are known as foci (singular focus).
- The major axis is the ellipse’s longest diameter (typically represented by ‘a’), which runs through the midpoint from one end to the other, at the broadest section of the ellipse. On the other hand, the minor axis is the ellipse’s shortest diameter (denoted by ‘b’), passing through the midpoint at its shortest end.
- The semi-major axis of an ellipse is half of its major axis in length.
Quiz
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Frequently Asked Questions
If B and C are foci or the two stable ends encompassed by the curve. So, the ellipse semi major axis with A as any random end on the ellipse can be calculated as Semi Major Axis = (AB +AC)/2
OR
Semi-Major Axis = Half of the length of the Major Axis
An ellipse can be a perfect circle only when the major and minor axes are equal in length to each other.Â
Ellipses can be used in various real-world applications like engineering, astronomy, physics, architecture, and more.
When the major axis of an ellipse aligns with the polar line of the polar ellipse, and both the ellipses generate from a focus, we can refer to the situation as an ellipsed or dual ellipse. At the same time, the minor axis of the ellipse aligns with the axis of the polar ellipse.
Yes, there are different formulas for semi-major and semi-minor axes in an oblique ellipse.Â
