* P(r,theta)
/
/
/
r/
/<.
/ `.
/ theta\
/ .
O o-----+---'-+-----+-----> r
0 1 2 3
In this example, we have plotted the point (3,Pi/3).
Conversely, given a coordinate pair (r,theta), construct from pole O the ray making angle theta with the polar axis, and measure a distance of r along that ray. The endpoint is P.
Observe that (r,theta) and (-r, theta+Pi) are two different pairs of coordinates that represent the same point. Also observe that (0, theta) for any value of theta are infinitely many different pairs of coordinates that represent the same point, the pole O.
To transform from rectangular coordinates to polar ones and vice versa, use the following formulas:
x = r cos(theta),
y = r sin(theta),
r = ±sqrt(x2+y2),
theta = arctan(y/x).
The sign of r is determined by which of the values of the arctangent function is chosen:
Sign of x Sign of y Quadrant of theta Sign of r
+ + I +
+ + III –
– + II +
– + IV –
– – I –
– – III +
+ – II –
+ – IV +
The quadrant of theta can always be chosen to make r positive, if it is so desired.
Note: Many equations which are simple in Cartesian coordinates are very complicated in polar coordinates, and vice versa. To solve any particular problem, one system may be much more tractable than the other. The ability to convert from one to the other quite readily allows us to try both, then use only the easier one.
Let the coordinates of P1 be (r1,theta1), those of P2 be (r2,theta2), and those of P3 be (r3,theta3).
Points
The distance from P1 to P2 isd = sqrt(r12+r22-2r1r2cos[thet
The coordinates of the point dividing the line segment P1P2 in the ratio a/b are:(sqrt[b2r12+a2r22+2abr1r2c
As a special case, when a = b, the midpoint of the line segment has coordinates(sqrt[r12+r22+2r1r2cos(the
P1, P2, and P3 are collinear if and only if the determinant
r1cos(theta1) r1sin(theta1) 1
= 0,
r2cos(theta2) r2sin(theta2) 1
r3cos(theta3) r3sin(theta3) 1
or else
r1r2sin(theta2-theta1) + r2r3sin(theta3-theta2) + r3r1sin(theta1-theta3) = 0.
What is the point in arguing? Atheist and Christian apologetics (with regards to evolution or the existence of God) complicate the issue and confuse those who want practical layman answers. There are brilliant minds on BOTH sides of the argument. Yet brilliant minds can not concede or agree that one or the other is correct? So who is right?
Art: Marco Brambilla |
Art: Edwin Lester |
mathematical genius. Give me a break! I am a stupid creature!
I have concluded that there is nothing, per modern day, that can be used to prove God’s existence. Nothing, per atheist standards. It would take an apple falling from a tree so to speak. What does this amount to modern day? The rapture of God’s people. WHEN this event occurs, atheists will witness unequivocal evidence that God exists. The burden is now on me to prove that the rapture will occur. It’s a never ending battle.
This argument is simple to me, and should be simple to anyone else who chooses to investigate. Christianity is based in faith on the words of the Bible. We believe that EVERYTHING created, has a purpose and its construction was initiated by a superior and intelligent creator. As an American, I have the right to believe foolishly (according to Atheists and anti-Christians), wisely (according to fellow believers), possibly (according to Agnostics), and indifferently (to those who could care less). I shouldn’t be suppressed in the public/national arena or invalidated/ostracized for my belief, nor should I force or impose my belief on anyone else.
I refuse to question my faith because by doing so, I have already failed at its purpose. I fail enough every day as it is. We can all learn valuable lessons from everything/everyone around us and we can use that knowledge to boost our respect and understanding of other cultures, religions, and people. Redefining our faith, however, based on external lessons learned and application through personal selfishness, to me, seems like a recipe for disaster.
This is my point. Believe in what you will! If you are an atheist and someone asks you to prove God doesn’t exist, then by all means, provide them with books, science formulas, lunar equations, and testimony from brilliant minds showing that without a doubt, God is a fairytale.
Let me have the right, however, to tell someone that God created the earth in seven days. Mankind chose free will over eternal paradise. God sent his Son to die a gruesome death and paid for our sins with His blood. He rose from the dead, and He’s gonna be back soon.
That’s my right. And I am in my right mind….
is that what your book will be about? if so, im absolutely interested
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