Polar Graphing Project
Polar Graphing ProjectInvestigating Various Types of Polar GraphsCardioidThe cardioid polar graph is a special case of Pascal’s limacon. The term cardioid was first used by Johann Castillon, an Italian astronomer and mathematician. However, Philippe de la Hire, a French mathematician, is known as the discoverer of the cardioid. La Hire found the length of the cardioid in 1708. Because of its relation to the limacon, study of this graph first occurred long before the time of Castillon or la Hire. The word itself comes from the Greek kardio, meaning heart, referring to the shape of the graph. The cardioid graph can be shown with the formula r=a±a(cosθ) or r=a±a(sinθ). As a general statement, the difference in graphs using sine and cosine is that sine graphs are a rotation of 90? from the cosine graphs. In Case I, where a is an element of the integers and a is greater than or equal to zero, a changes the overall size of the curve. Whether even or odd, the size of the graph increases as a increases. The value of a dilates the cardioid graph. Figure 1. Cardioid Cosine Graph with an Even ValueFigure 1 shows a cardiod cosine graph with equation r=4+4cos?(θ). When adding cos(?), the graph lies to the right of the y-axis. The diameter of this curve is 8, which is twice the value of a. Note: All graphs in this paper were created using TI-Nspire Student Software. Figure 2. Cardioid Sine Graph with an Odd ValueFigure 2 shows a cardioid sine graph with an odd a value. This graph has an equation of r=3-3sin(θ). When subtracting sine, the graph falls below the x-axis. The diameter of this curve is 6, which is two times the value of a.In Case II, where a is an element of the integers and a is less than zero, a also changes the size of the curve. Whether the value of a is even or odd does not seem to have an effect, as the graph is still dilated. As a decreases, the overall size of the curve increases, causing a dilation.Figure 3. Cardioid Cosine Graph with a Negative Odd ValueFigure 3 shows a cardioid cosine graph with a negative odd value for a. The equation that represents this graph is r=-3+3cos(θ). The diameter is 6, which is two times the value of a. Even though the value of a is negative, there isn’t much effect on the graph. Figure 4. Cardioid Sine Graph with a Negative Even ValueFigure 4 above shows a cardioid sine graph with a negative even value for a. The equation that represents this graph is r=-2+2sin(θ). The diameter is 4, which is two times the value of a. Even though the value of a is negative, there isn’t much effect on the graph. In Case III, where a is an element of the real numbers and a is not an integer, a does not seem to have any major effects on the cardioid graph. Whether the values of a are rational or irrational, the curve is dilated as a approaches infinity.Figure 5. Cardioid Cosine Graph with a Rational ValueFigure 5 above shows a cardioid cosine graph with equation r=4.5-4.5cos(θ). The diameter of this graph is 9, which is two times the value of a. Even though a is rational, it still has the same effect on the curve as an integer would. Figure 6. Cardioid Sine Graph with an Irrational ValueFigure 6 above shows a cardiod sine graph with an irrational a value of π The equation that represents this graph is r=π+πsin(θ). The diameter of this curve is twice the value of π. The irrational value of a in this graph has the same effect as an integer value would.LituusThe Lituus curve was discovered by mathematician Roger Cotes in 1722. The word Lituus means “a curved staff used in quartering the heavens.” The graph of the Lituus is a spiral described by the polar equation r= a2θ. As θ approaches infinity, the curve of the graph approaches the origin. An interesting fact about the Lituus spiral is that it is a recurring shape in the art form of Volute, often seen in architecture or the creation of musical instruments. In Case I, where a is an element of the integers and a is greater than or equal to zero, a changes the overall size of the curve. Whether even or odd, the size of the graph increases as a approaches infinity. A dilation is caused by a.Figure 7. Lituus Graph with Even ValueFigure 7 above shows a Lituus curve with equation r= 42θ. The window shown is standard, but the axes were shifted to show the entire curve. The value of a affected the graph by dilating the curve. Figure 8. Lituus Graph with an Odd ValueFigure 8 shows a graph of a Lituus Curve with an odd value as a. This graph has an equation of r= 32θ. The window is standard with shifted axes to show the entire curve. The value of a affects the graph by dilating the curve. In Case II, where a is an element of the integers and a is less than zero, a also changes the size of the curve. Whether the value of a is even or odd does not seem to have an effect. As a approaches negative infinity, the overall size of the curve increases, causing a dilation. Figure 9. Lituus Curve with a Negative Even ValueFigure 9 shows a Lituus graph with a negative even value for a. This graph has an equation of r= -22θ. The negative sign does not seem to affect the graph at all. The window of this graph is standard with the axes shifted. The a value dilated the graph. Figure 10. Lituus Curve with a Negative Odd ValueFigure 10 above shows a Lituus curve with equation r= -52θ. The window is standard, though the axes are shifted to show the whole graph. The a value in the equation dilates the curve. The negative value did not seem to have much effect. In Case III, where a is an element of the real numbers and a is not an integer, a does not seem to have any major effects on the curve. Whether the values of a are rational or irrational, the curve is dilated as a approaches infinity. Figure 11. Lituus Curve with a Rational ValueFigure 11 shows a Lituus graph with equation r= 2.52θ. The value of a diluted the curve. The window is standard with shifted axes to see the whole curve. The fact that the a value was rational did not have much effect on the curve; it was still dilated, though by a rational value.Figure 12. Lituus Graph with an Irrational ValueFigure 12 above shows a Lituus curve with an irrational value for a. The equation for this graph is r= π2θ. The irrational value did not have much effect on the curve, as the a value still dilated the curve. The window is standard with axes shifted to show the entire curve. Real-World ApplicationsPolar graphing goes beyond the classroom. There are many real-world uses of polar graphing out there. Polar graphs and coordinates are used in situations where it would be difficult to use rectangular graphs or coordinates, such as applications involving circular geometry. Because polar graphs are more circular or curved than most rectangular graphs, polar coordinates are often more useful in replicating models of things in the real world. Polar graphing is often used in modeling phenomena in space. For example, polar coordinates can be a model of a planet orbiting around the sun. Polar coordinates r and ? are used to show the radial distance between the sun and a given planet and the angular bearing of the sun to the planet respectively. Using these pieces of information, scientists are able to determine planetary orbitals in outer space. Polar graphing plays a major role in creating different types of microphones. Depending on the purpose of the microphone, there may be a different recording pattern. One type is an omni-directional microphone, which is used when recording from many different directions. This is useful in recording choirs or orchestras, for example. The recording pattern of this microphone is nearly circular. This goes along with the polar curve r=sin(?). This type of microphone has the ability to pick up on many different frequencies at once. A second type of microphone that utilizes the properties of polar graphs is a cardioid microphone. This microphone is uni-directional, meaning it only picks up sound from one direction. The recording pattern for this type is a cardioid, with an equation r=1+sin(?). Because the shape of the cardioid is not perfectly circular, there is what is known as “spill,” where some sounds behind the microphone are picked up on. Other types of microphones that use polar graphing are “super-directional” shotgun microphones and bi-directional microphones. The shotgun microphone picks up on sound only directly in front of it, so it is used in situations where extra noise should be eliminated. The graph that represents this microphone is r=?2. Bi-directional microphones are used mainly in interviewing situations where the voices of both people need to be heard. The equation that represents this is r=cos(?)2. Microphones that use polar graphs are not only professional studio-quality types. A person could be coming into contact with one every day without even realizing it. Laptops, cellphones, iPods, and cameras all have microphones that are based off of polar graphs. In today’s technological society, polar graphing plays a little known daily role. Polar graphing may play a prominent part in classrooms and laboratories, but there are true “real-world” applications of it. BibliographyBourne, Murray. "SquareCircleZ." Polar Coordinates and Cardioid Microphones ::. SquareCircleZ, 18 Oct. 2009. Web. 05 May 2012. <, Jeffrey L. "What Are Polar Coordinates?" WiseGeek. Ed. A. Joseph. Conjecture. Web. 05 May 2012. <, Richard. "Planetary Orbits." Planetary Orbits. University of Texas at Austin, 2 Feb. 2006. Web. 05 May 2012. <, Xah. "Lituus." Special Plane Curves. Differential Equations, Mechanics, and Computation. Web. 01 May 2012. <'Connor, John J. "Cardioid." MacTutor History of Mathematics. University of St. Andrews Scotland, Jan. 1997. Web. 01 May 2012. <;. ................
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