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Start with a point O in the plane, the pole. Through the pole O choose a ray (half a line) bounded by O. This is the polar axis. To any point P corresponds a pair of real numbers called its polar coordinates, r and theta, determined as follows. Connect P to O. Measure the distance r from pole O to point P, and measure the angle theta between the axis and OP in a counterclockwise direction in radians (1 radian = 180/Pi degrees). Here is a diagram to illustrate this:
* 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[theta2-theta1]).
The coordinates of the point dividing the line segment P1P2 in the ratio a/b are:(sqrt[b2r12+a2r22+2abr1r2cos(theta2-theta1)]/[a+b], arctan([br1sin(theta1)+ar2sin(theta2)]/[br1cos(theta1)+ar2cos(theta2)]).
As a special case, when a = b, the midpoint of the line segment has coordinates(sqrt[r12+r22+2r1r2cos(theta2-theta1)]/2, arctan([r1sin(theta1)+r2sin(theta2)]/[r1cos(theta1)+r2cos(theta2)]).
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.
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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?
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| Art: Marco Brambilla |
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| Art: Edwin Lester |
Atheists, essentially, want every Christian to solve the polar coordinate math problem seen above. If you can’t solve the problem, then you do not have an argument to prove to that person that God really exists. Some Atheists even go so far as to say that if you can’t solve this math problem (hypothetical), yet you believe the answer given by a mathematical genius, then you are an idiot for believing the genius and deceived for not figuring out the math problem yourself. The burden is now on me to prove that the mathematical genius is ACTUALLY a
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….
Rhetoric resonates within our society and turns many sheepish heads in directions they may have never turned if it hadn't been for disaster. You might be used to rhetoric, babe, but I am presenting something that you are not accustomed to. I have already used empty words and they were cast into a vault in Las Vegas. There they lay. They corrode the steel that enslaves them. They pollute those who come within breathing distance. They pollute me no more.
Normalcy teases me like you do. Quit playing to the wishes and desires of others and fashion your life around the things that provoke the life within YOU. The world grows evil around us, and I fear our time is short. Better to have loved then to have never taken the chance. Maybe my years of experience and failure have engrained a yearning for monotony with a splash of your adventure. Will you choose to season me further? Or will you leave me bland based on your desire to please everyone but yourself?
Love waits for you behind a door that you refuse to open. Granted, a snake laid in wait behind the rusty black door you opened last. You were bitten and laid paralyzed in a cloud of confusion and uncertainty.
Awake! The antidote has been placed at your feet and you stare at it devoid of belief. Full relief will not take affect by sipping a drop at a time. Consume the antidote in full and welcome yourself to a new world as cured! Like everything else, the antidote has an expiration date. It will sour with time and eventually will turn to a poison far deadlier then that which already circulates through your veins.
A new door has been placed in your path. Open it…Your cure will stand as complete…
I asked for a splash of humiliation and I received a handful of ash. Keep in mind that ash cures as charcoal, mostly the color of the average human heart. Can it catch flame and hold fire? I have witnessed this. I have been impressed at times and others I have been horrified! So what is my drink of choice this fine September eve? I'll take a twist of poison and drink willingly. I'll make sure to drown the cure into the fellow sitting in the third barstool to the left of me. He seems so dreary, yet strikingly alive. His child just graduated from college, a promotion fell his way. But his wife left him for a lawyer, said he could not satisfy her with his current pay. From what I understand, he toiled for years and years. He suffered through the sorrow while he fought back the tears. I'll drink to those tears and I'll share with him more. I'll be the shoulder that he would need right now, more then ever before.
I find it strange that she cannot appreciate, the strength that he shows during times that are hard, and going that is rough. He puts up with your pain, he can't get enough. If she places strange skin on a pedestal, high above the wrinkles that she has become used to, how in the world does she think she will survive when hard times come without a binding contract? He has nothing to lose. Except for you, because you are binding, to someone else. And you will lose everything that you said that you love. The gutter will accept you with open arms, and the rain from clouds and eyes will keep you cold on this cold night. The only warmth you will find is orange and wrapped in cylindrical glass, and even she will make you sick, faster then you have ever become before. Find solace in the fact that you will be denied change willingly, for you have changed unwillingly. Blame it on the needle and actions of the past. Your days will be long, you're numbered, you're last…in line at the clinic to cure you, the disease… You shake like you've been broken. Are you kidding me? You haven't seen nothing! You could have avoided your current situation. Instead you embraced it and welcomed it with forced open arms. It's your fault that you died today….
I am dazed. Take it from me; a tornado has no mercy when it kills you. I was killed by fierce wind and lived to tell the tale. The remnants of a monster lay strewn across three states, thankfully. I was ripped to pieces, and when the dust settled I was able to examine each part of me that I could find.
I found my heart in Nevada. It was black, hard, and cold. It was torn to pieces. Not by the wind, but by the person that held it before tragedy. It was beating but barely.
I found my head in Colorado. My head holds my mind, but for some reason even THAT was lost somewhere in the clouds. My head tells me that it was in cahoots with my heart. They worked in unison to destroy my body. They succeeded, but to what avail? In the process, head and heart are lying in the woodsy state where they fell.
I found my hands in Southern California. They were not calloused or bloodied. They were not calloused and bloodied because they functioned improperly. They did not hold, they did not work, and they did not love correctly. They established a relationship with the pockets in my jeans and they slept tucked away from responsibility.
I gathered all of the pieces that I could find and bought some glue, nails, and duct tape. I took my heart and wiped away the Nevada sand. I gave it to God and he dusted off the black. He cupped it in his hands and made it warm, which in turn softened it to factory texture. Then he blew into one of the valves and it pumped with rapid new life. I placed the heart into the hole in my chest and covered the hole with duct tape.
I took my head and placed it on my neck. I fastened it securely with nails. I called a conference and asked mind and heart to attend. The two were able to come to a truce and now work in unison to achieve a common goal. My mouth is now able to talk. My mouth can smile for no reason. My eyes are no longer cast to the ground. They look straight ahead. My eyes shine the color blue proudly. Apparently, everything is in working order now. And I believe my head! For the first time in a little over a year, I actually believe my head!
I took my hands and glued them to my wrists. If I could do it with glue and no experience, why can't doctors do it with surgical tools and experience? My hands hate my pockets now. They want to hold, feel, rise to the sky, and they want to love. Never again will they be idle.
The monster is gone. He existed, but has been destroyed. He isn't hiding in the closet or under my bed. He doesn't control my mind or actions any longer. In its place stands a new, old man. I am tattered, a little wrinkled, and a little tired, but I am overjoyed because I live. And I have so much to live for….
So quick to make judgments! I am usually strong but am vulnerable to your poison at times. I've wronged and been forgiven yet you still insist on giving me hell for something you had no part or place in. You were quick to turn my cheek and then you default on your mortgage because you HAD to buy the coolest new pocket pleasure. Don't cast hatred when you can't control impurities within your own life. So turn my other cheek. I am giving it to you willingly. Your hypocrisy is blinding to me and has obviously rendered you ignorant.
I don't drink much. I don't fly on anything. I find elation in doing things that may seem meaningless. How can you make the assumption that I am not worth the time without trying me out? You ever think you might like taking it easy for a change? I've already lived fifty years in this thirty-year old body. I don't need to prove anything to you. I don't need to swallow Absolut or Patron in excess to have a good time. I already did that. When all was said and done, all I possessed was an aching body and a sickened internal makeup. You are young. When you are my age, you will pass on the same knowledge and agree with my words wholeheartedly.
So open your eyes. Make judgments based on character as it relates to adding someone new to your life. Drunken nights will fade your years and you'll wake up wondering what happened. Your rejection of the caring people will have blown away like ashes in the wind and you will curse the day you let the solid objects go. For the solid objects root you, and the liquid you worshipped will seep into the ground and produce withered weeds. Weeds will serve as a reminder of the sail you tore when you had the option to, instead, let it stay whole so the wind could fill its back. The only reason you haven't joined the soil is because you are beautiful, and beauty will always be spared.
This is a general statement.
