Plane Geometry Exploration of 
Sir Isaac Newton's Universal
Inverse Square Law of Gravity

G-Field Force(acceler) ∝ G((((Subscript[M, 1]) × (Subscript[M, 2]))/(r^2)))

Wolfram Technology Conference 2014

ON CONSTRUCTING MECHANICAL ENERGY 
CURVES OF GRAVITY.

ALEXANDER; CEO SANDBOXGEOMETRY LLC.

constructing a curved space division assembly

lets construct Euclidean Tangents at the north and south spin axis of our traditional unit circle between (0, 1) and (0, -1).

ParametricPlot[{{Cos[t],Sin[t]},{t,1},{t,-1}},{t,0,6π},PlotRange->{{0,3},{((-3)/2),2}},AxesOrigin->{0,0}]

Changing Euclidean objects into curved space Euclidean objects of space and time?Add some Gfield mass/volume ratio making FIELD POTENTIAL summed content of the unit circle.

((t^2)/(∓4(p)))±r ⟶ dependent time &  energy curve (r = p)

{1Cos[t],1Sin[t] ⟶ independent gfield potential curve

ASSIGN FIELD POTENTIAL TO (UNIT CIRCLE) INDEPENDENT CURVE

ParametricPlot[{{Cos[t],Sin[t]},{t,((t^2)/(-4))+1}},{t,-π,2π},PlotRange->{{-2,3},{-1,1}}]

BASIC GRAVITY FIELD ORBITAL (CSDA)

{rCos[t],rSin[t]},{t,((t^2)/(-4(p)))+r}

Using a CSDA we can go from our 450 year old popular perception of gfield motion given us by Galileo, Kepler, and Sir Isaac Newton...

to 21st century perception of gfield spin (range) and accretion rotation (domain) using Computer Algebra System (CAS).
Basic CSDA; spin and rotation map of (gravity field)(Subscript[M, 2]) and (Subscript[M, 1])

CDF of CSDA:Code by Abraham Gadalla
A COMPUTER DOCUMENT FORMAT DEMONSTRATING CHANGING ORBIT ENERGY (f (r)) FOLLOWING AN EVENT RADIUS (r).

Reading KE and momentum using a CSDA space and time curve

∫(m(v))\[DifferentialD]v == ((m (v^2))/2)

(∂_v)((1/2)m((v^2))) == m v

Part 2: 
CONSTRUCTING SPACE AND TIME

Lets construct a space and time square for our earth moon system.

BUILDING EARTH AND MOON SPACE AND TIME SQUARE
NASA Moon/Earth Parameters

Orbital parameters (for lunar orbit about the Earth)

Semimajor axis          384400 km
Perigee        	         363300 km
Apogee 	         405500 km

orbit KE in km/sec        
Mean orbital velocity (km/s)        1.022 km/s           
Max. orbital velocity (km/s)        1.076 km/s
Min. orbital velocity (km/s)         0 .964 km/s

converting orbit parameter into CSDA unit meter

({{moon, NASAparameters, conversion, CSDA parameters}, {perigee, 363300, (363300/384400)*2, 1.8902}, {apogee, 405500, (405500/384400)*2, 2.1098}, {AVERAGE, 384400, (384400/384400)*2, 2}, {potential, (r=p=(aver/2)), ((384400/2)/(384400/2)) *1, 1}, {f(p), 20521, \[Placeholder], 0.1068}, {f(a), -21680, \[Placeholder], -0.1128}, {focalradius(p), 363879, 2r-f(r), 1.8932}, {focalradius (a), 406080, 2r-f(r), 2.1128}})

FINDING ENERGY LEVEL: ((t^2)/(-4(p)))+r/.t->event radius

FINDING FOCAL MAG (2p - f(r)) = focal radius

Step1: Construct a basic CSDA

CURVED SPACE DIVISION ASSEMBLY

Step 2: construct space and time square; using event radii limits and energy level limits. 
High energy coordinates are red; low energy coordinates are blue.

ParametricPlot[{... {1.8902, t}, {2.1098 t}, {t, 0.1068}, {t, (-0.1128)}} ...]

Establish Mathematica template for energy level:

((t^2)/(-4))+1/.t->1.8902->(.1068)energy level

((t^2)/(-4))+1/.t->2.1098->(-0.1128)energy level

Step 3:  focal properties of curved space

ParametricPlot...[{{t,-0.1128-0.0535 (-2.1098+t)},{t,0.1068+0.0565(-1.8902+t)}}]

[y-energy=((energy)/(eventradius))(x-eventradius),y]

event radius = 1.8902; and event energy = 0.1068 ;  event radius = 2.1098; and event energy = -0.1128

Solve[y-(-0.1128)==(((-0.1128))/2.1098)(x-2.1098),y] →{y->-0.1128-0.0535 (-2.1098+t)}

Solve[y-0.1068==(0.1068/1.8902)(x-1.8902),y] → {y->0.1068+0.0565(-1.8902+t)}

magnitude of curved space focal radii

2(p) - f (r) = focal radius magnitude; [2 - (orbit energy level) = (focal radii magnitude)].

[2-(-0.1128)→ 2.1128] and [2- 0.1068 →  1.8932]

Find size of focal radius to orbit parameter limits (2p - f (r) = magnitude) :

HIGH ENERGY focal property r = 1.8932         event r = 1.8902  
(increase 30 pts)

LOW ENERGY focal property r =2.1128;          event r = 2.1098   
(increase  30 pts)

FINDING (HIGH ENERGY) ANGULAR MOMENTUM

Event radius = 1.8902;  focal radius = 1.8933
NASA HIGH ENERGY MOMENTUM = 1.076 KM/SEC

(∂_t)(((t^2)/(-4(p)))+r) where (p) and (r) = 1 becomes (((-t)/2))

Solve[(((m=1))/1.022)==(((((1.893/2)))^(-1))/v),v] ⟶ 1.080 
1.080 - 1.076 → 0.004 (difference of 4 meters/sec FASTER).

finding momentum values of energy tangents (LOW ENERGY BLUE)

MIN ORBIT VELOCITY = 0.964 km/sec    FOCAL RADIUS MAG = 2.1128

NASA: Min. orbital velocity (km/s) 0 .964 km/s

Solve[(((m =1))/1.022)==(((((2.1128/2)))^(-1))/v),v] ⟶ 0.969 km/sec

.969-.964 → 0.005 (difference of 5 meters/sec FASTER).

PART 3:Inverse square law and Curved Space Plane Geometry

We are now ready to prove event radii shape the motive curve of the gravity field using Sir Isaac Newton' s Inverse Sq Law; I will do so with unity curves
(curvature and radius of curvature = 1)

PROPOSAL: LET THERE BE TWO CURVES COMPOSING AVAILABLE ENERGY EXCHANGE BETWEEN POTENTIAL AND MOTION ((Subscript[M, 1]) ⟺ (Subscript[M, 2])):

THEOREM: LAW OF CONSERVED ENERGY AND G-FIELD ORBIT MOTION: Since energy exchanged between these two curves (motion and potential) determines orbit momentum, we need two equal curves to initialize shared energy QUANTITY; when added together zero balance the exchange for stable orbit motion. Somewhere, on the period time curve, there will be a motive curve of same shape as potential less the content.  Enter the latus rectum average orbit diameter, reference level of gravity field orbit energy curves. It is here, and only here, on the average diameter of an orbit can two unity curves co-exist.

Construct two unity curves, one as central force potential, and one at energy tangent (slope - 1) event on gfield time curve.

POTENTIAL IS COLLECTION OF MICRO INFINITY CURVATURE; MACRO INFINITY RADIUS OF CURVATURE IS OUR SQUARE SPACE

Going from orbit radius of curvature to inverse square event curvature. [({{(Subscript[F, acc])∝G, (((Subscript[M, 1])(Subscript[M, 2]))/(r^2));   ∝(k×((((1/r)))^2))}})]

To use Sir Isaac Newton’s Inverse Square law to construct and prove shape of energy curves, I need to roll the fixed parameters of event radius orbit math into one constant of proportionality as coefficient to curvature.

since (Subscript[M, 1]) and (Subscript[M, 2]) won't change much in our lifetime roll them into G and let them = constant of proportionality k; we have:
(k×((((1/r)))^2))

to surmise the shape of the motive energy curve we invert. I will show the CSDA latus rectum diameter is the g-field constant of proportionality.

(((k×((((1/r)))^2)))^(-1)) = shape of motive energy

Prove shape of average motive curve = shape of potential curve.
1. Construct reach of gfield potential as a tangent limit through orbit space. (t, 1)
           2. Construct shape of potential curve (curvature = 1) about central force F; given. (Cos[t], Sin[t])
      3. Compute and construct shape of motive curve at event energy slope (m = -1), using
a) focal property differential; 
	                           b) Sir Isaac Newton inverse square law.

a)  radius of motive curve = (focal radius mag - potential) ⟶  2 - 1 = 1 unit

b) radius of motive curve using inverse square-> [(((((((1/2)))^2)×4(1)))^(-1))⟶1 unit]

QED; (curvature of potential curve = 1); and (curvature of motive curve + 𝒻 (r) = 1).

CONSTRUCT HIGH ENERGY MOTIVE CURVE OF EARTH MOON: EVENT RADIUS: 1.8902; FOCAL RADIUS:1.8932;

Substract (potential) from focal radius; (1.8932 - 1 = .8932); motive curve radius = .8932

{.8932Cos[t]+1.8902, .8932Sin[t]+0.1068}

Sir Isaac Newton's Inverse Sq Law: event radius = 1.8902
shape of motive curve =(((((((1/1.8902)))^2)*4))^(-1))⟶ 0.893214

{ParametricPlot... {0.8932 Cos[t] +1.8902, 0.8932 Sin[t] + 0.1068} ...

CONSTRUCT LOW ENERGY MOTIVE CURVE OF EARTH MOON

EVENT RADIUS: 2.1098; FOCAL RADIUS:2.1128 -1 = 1.1128 motive curve;

Sir Isaac:(((((((1/2.1098)))^2)*4))^(-1))=1.1128

ParametricPlot[...{2.1098,t},{t,-0.1128},{1.1128Cos[t]+2.1098,1.1128Sin[t]-0.1128}...]

INSULATOR TANGENTS; SEPARATING OPPOSING FORCES OF ATTRACTION AND ESCAPE.

Construct High Energy Insulator Tangent; focal slope =  (0.1068/1.8902); fr = 1.8932.
SLOPE OF INSULATOR IS NORMAL WITH THE FOCALRADIUS

compute abscissa of insulator tangent ⟶ Solve[(1/x)==(1.8932/1.8902),x] → {x->0.9984}

compute ordinate of insulator tangent ⟶ Solve[(1/y)==(1.8932/0.1068),y] →  {x->0.0564}

Solve[y-0.0564==(((((-.1068)/1.8902)))^(-1))(x-0.9984),y] ⟶{t,0.0564 -17.6985 (-0.9984+t)}

INSULATOR TANGENTS; SEPARATING OPPOSING FORCES OF ATTRACTION AND ESCAPE.

Construct Low Energy Insulator Tangent; focal slope =  (((-0.1128))/2.1098); fr = 2.1128.
SLOPE OF INSULATOR IS NORMAL WITH THE FOCALRADIUS

insulating tangent  ordinate:        Solve[(1/y)==(2.1128/(-0.1128)),y] ⟶  y->-0.0534

insulating tangent abscissa ⟶   Solve[(1/x)==(2.1128/2.1098),x] ⟶  {x->0.9986}

Solve[y-(ordinnate)=changesign(((focalslope))^(-1))(x-abscissa)=

{{y->-0.0534+18.7037 (-0.9986+x)}} = {t,-0.0534+18.7037 (-0.9986+t)}

CONVERGENCE POINT OF MOTION ENERGY AND ANGULAR MOMENTUM (B): 
All energy tangent phenomena (motive and insulator) connect at curved space directrix at point B (event radius/2).

Motive energy curves change shape to accommodate conserved angular momentum  experienced by changing orbit event radii.

ALL TANGENT INTERCEPTS = EVENT RADIUS/2

high energy event radius = 1.8902

high energy event tangent = {t,0.1068 -0.9451 (-1.8902+t)

high energy insulator = {t,0.0564 -17.6985 (-0.9984+t)

Solve[0.0564 -17.6985 (-0.9984+t)==0.1068 -0.9451 (-1.8902+t),t]

(1.8902/2) -> 0.9451

low energy event radius: 2.1098

low energy event tangent: {t,-0.1128-1.0549 (-2.1098+t)}

low energy insulator:  {t,-0.0534+18.7037 (-0.9986+t)}

Solve[-0.1128-1.0549 (-2.1098+t)==-0.0534+18.7037 (-0.9986+t),t]

(2.1098/2) -> 1.0549

COPYRIGHT ORIGINAL GEOMETRY BY: 
Sand Box Geometry LLC, a company dedicated to utility of Ancient Greek Geometry in pursuing exploration and discovery of Central Force Field Curves.

finding an Apollonian parabola section focus (cone slope m = ± 2;  vertex @ unit cir dia):

THANK YOU

Alexander@sandboxgeometry.com