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Figure 10-7 Illustration for Question and Answer 9-1
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The height h can be any real number There are no restrictions on it whatsoever We have h > 0 if and only if P is above the reference plane, h < 0 if and only if P is below the reference plane, and h = 0 if and only if P is in the reference plane
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Question 9-3
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Consider a point P = (q,r,h) in cylindrical coordinates How can we determine the coordinates of P in Cartesian xyz space
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The Cartesian x value of P is x = r cos q The Cartesian y value is y = r sin q The Cartesian z value is z=h
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Question 9-4
Consider a point P = (x,y,z) in Cartesian three-space How can we find the direction angle q of the point P in cylindrical coordinates
Answer 9-4
Cartesian-to-cylindrical angle conversion is the same as Cartesian-to-polar angle conversion: If x = 0 and y = 0, then q = 0 by default If x > 0 and y = 0, then q = 0 If x > 0 and y > 0, then q = Arctan ( y /x) If x = 0 and y > 0, then q = p /2 If x < 0 and y > 0, then q = p + Arctan ( y /x) If x < 0 and y = 0, then q = p If x < 0 and y < 0, then q = p + Arctan ( y /x) If x = 0 and y < 0, then q = 3p /2 If x > 0 and y < 0, then q = 2p + Arctan ( y /x)
Question 9-5
Consider a point P = (x,y,z) in Cartesian three-space How can we find the radius r of the point P in cylindrical coordinates How can we find the height h of the point P in cylindrical coordinates
Part One 205 Answer 9-5
To find the cylindrical radius coordinate of P, we find the distance between its projection point P and the origin in the xy plane The z value is irrelevant, so the formula is r = (x2 + y2)1/2 The cylindrical height is simply equal to z The x and y values are irrelevant, so the formula is h=z
Question 9-6
How do we determine the spherical coordinates of a point in three-space
Answer 9-6
We start with a Cartesian reference plane The positive Cartesian x axis forms the reference axis To determine the spherical coordinates of a point P, we first locate its projection point, P , on the reference plane: The horizontal angle q turns counterclockwise in the reference plane from the reference axis to the ray that goes out from the origin through P The vertical angle f turns downward from the vertical axis to the ray that goes out from the origin through P The radius r is the straight-line distance from the origin to P The basic scheme is shown in Fig 10-8 We express the spherical coordinates as an ordered triple P = (q,f,r)
+z Reference axis
P r x P
Reference plane
Figure 10-8 Illustration for Question and
Answer 9-6
Review Questions and Answers Question 9-7
Are their any restrictions on the horizontal or vertical angles in spherical coordinates Are there any restrictions on the radius
Answer 9-7
Theoretically, we can have a nonstandard horizontal direction angle, but it s best to add or subtract whatever multiple of 2p will bring it into the preferred range 0 q < 2p If q 2p, it represents at least one complete counterclockwise rotation from the reference axis If q < 0, it represents clockwise rotation from the reference axis Theoretically, we can have a nonstandard vertical angle, but it s best to restrict it to the range 0 f p We can do that by making sure that we traverse the smallest possible angle between the positive z axis and the ray connecting the origin with P The radius can be any real number, but things are simplest if we keep it nonnegative If we find ourselves working with a negative radius, we should reverse the direction by adding or subtracting p to or from both angles, making sure that we end up with 0 q < 2p and 0 f p Then we must take the absolute value of the negative radius and use it as the radius coordinate
Question 9-8
Consider a point P = (q,f,r) in spherical coordinates How can we determine the coordinates of P in Cartesian xyz space
Answer 9-8
The Cartesian x value of P is x = r sin f cos q The Cartesian y value is y = r sin f sin q The Cartesian z value is z = r cos f
Question 9-9
Consider a point P = (x,y,z) in Cartesian three-space How can we find the horizontal angle coordinate q of the point P in spherical coordinates
Answer 9-9
The Cartesian-to-spherical horizontal-angle conversion process is identical to the Cartesianto-cylindrical direction-angle conversion process: If x = 0 and y = 0, then q = 0 by default If x > 0 and y = 0, then q = 0 If x > 0 and y > 0, then q = Arctan ( y /x)
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