Website owner: James Miller
CHANGES IN EQUATION COEFFICIENTS PRODUCED BY TRANSLATION AND ROTATION OF THE COORDINATE SYSTEM
Changes in equation coefficients produced by translation. What changes are produced in the coefficients of the general equation of the second degree
1) f(x, y, z) = ax2 + by2 + cz2 + 2fyz + 2gxz + 2hxy + 2px + 2qy + 2rz + d = 0
by a translation of the coordinate system? Let x'-y'-z' be a coordinate system whose origin is located at point (x0, y0, z0) of an x-y-z system with corresponding axes parallel. Let us observe the result of substituting
x = x' + x0
y = y' + y0
z = z' + z0
into equation 1) above.
The equation becomes
f(x', y', z') = f(x' + x0, y' + y0, z' + z0) = a(x' + x0)2 + b(y' + y0)2 + c(z' + c0)2 + f(y' + y0)(z' + z0) + 2g (x' + x0)(z' + z0) + 2h(x' + x0)(y' + y0) + 2 p(x' + x0) + 2q (y' + y0) + 2r(z' + z0) + d
= ax' 2 + by' 2 + cz' 2 + 2fy'z' + 2gx'z' + 2hx'y' + 2(ax0 + hy0 + gz0 + p)x' + 2(hx0 + by0 + fz0 + q)y' + 2(gx0 + fy0 + cz0 + r)z' + ax02 + by02 + cz02 + 2f y0z0 + 2gx0z0 + 2hx0y0 + 2px0 + 2qy0 + 2rz0 + d
= ax' 2 + by' 2 + cz' 2 + 2fy'z' + 2gx'z' + 2hx'y' + 2(ax0 + hy0 + gz0 + p)x' + 2(hx0 + by0 + fz0 + q)y' + 2(gx0 + fy0 + cz0 + r)z' + f(x0, y0, z0) = 0
We see that a translation has no effect on the coefficients of the second degree terms (i.e. the coefficients a, b, c, f, g, h). It only changes the first degree coefficients p, q, r and the constant term d where
p ax0 + hy0 + gz0 + p
q hx0 + by0 + fz0 + q
r gx0 + fy0 + cz0 + r
d f(x0, y0, z0)
Changes in equation coefficients produced
by rotation of the coordinate system
about it origin. Let the origin of a x'-y'-z' system be coincident with the origin of an x-y-z system and let λ1, μ1, ν1 be the direction cosines of the x' axis, λ2, μ2, ν2 be the direction cosines of the y' axis, λ3, μ3, ν3 be the direction cosines of the z' axis as referred to the x-y-z system.
x = λ1x' + λ2y' + λ3z'
y = μ1x' + μ2y' + μ3z'
z = ν1x' + ν2y' + ν3z'
Question. What effect does this rotation of the coordinate system have on the coefficients of the general equation of the second degree 1)?
Answer. A rotation of the coordinate system causes changes in all the coefficients a, b, c, f, g, h, p, q, r. The constant term d, however, is unchanged.
The Way of Truth and Life
God's message to the world
Jesus Christ and His Teachings
Words of Wisdom
Way of enlightenment, wisdom, and understanding
Way of true Christianity
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Who will go to heaven?
The superior person
On faith and works
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
The teaching is:
On modern intellectualism
On Self-sufficient Country Living, Homesteading
Principles for Living Life
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
America has lost her way
The really big sins
Theory on the Formation of Character
You are what you eat
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
These things go together
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
The True Christian
What is true Christianity?
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Walking a solitary road
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
On responding to wrongs
Real Christian Faith
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
Sizing up people
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?