WOLFRAM TECHNOLOGY CONFERENCE

OCTOBER 2015

Taking Galileo, his Inclined Plane, Uniform Acceleration, and ‘Mean Speed Theorem’ to the 21st Century with Mathematica

Alexander CEO Sand Box Geometry LLC
alexander@sandboxgeometry.com

JMM 2015; SAN ANTONIO, TEXAS

ON CONSTRUCTING MECHANICAL ENERGY 
CURVES OF GRAVITY
 (M2 ORBITS).
 using machine technology

ALEXANDER; CEO SANDBOXGEOMETRY LLC.

Sand Box Geometry LLC, a company dedicated to utility of Ancient Greek Geometry in pursuing exploration, discovery and construction of Central Force Field Curves.

I have always been interested in utility of plane geometry to construct a standard model of gravity field mechanical energy curves. To do so I invented a Curved Space Division Assembly, acronym CSDA. A plane geometry construction combining two elementary curves, the unit circle and unit parabola.

WHAT IS A CURVED SPACE DIVISION ASSEMBLY?

+

+      =

note: (π/2) direction radius of unit circle is spin congruent with the vertex focal radius p of the unit parabola creating a plane geometry g-field function.

The Geography of gravity field CSDA plane geometry curves in terms of Calculus.

INDEPENDENT CURVE (OUR UNIT CIRCLE)

(1Cos[t], 1Sin[t]): A closed Constant  Acceleration surface curve used to study g-field potential and mechanical meter of “weight”(ie; surface acceleration phenomena of our earth).

DEPENDENT CURVE (SBG OPEN CURVE UNIT PARABOLA)

t, ((t^2)/(-4(p)))+r) Used to meter Changing Acceleration phenomena; altered kinetic energy that curves orbit motion.

Together, using one shared point as center and focus, these two curves become:

A plane geometry function capable of analyzing gravity field phenomena of constant and changing acceleration forces as well as mechanical meter of required energy for seperation between field potential (M1) and motion (M2).

Assigns curve heirarchy using two principal properties of curved space.{position, and position energy} 

Combined, these two curve compose a CSDA; Curved Space Division Assembly.

To explore uniform (constant) acceleration phenomena we begin with GALILEO and his INCLINE PLANE.

Giuseppe Bezzuoli (28 November 1784 – 13 September 1855) was an Italian painter (FROM WIKEPEDIA)

By calibrating beats of time with roll space down an incline plane, Galileo found that fall rate (uniform acceleration) is a series sum of odd numbers directly proportional to the square of time. 
http://www.mcm.edu/academic/galileo/ars/arshtml/mathofmotion1.html.

Analyzing Galileo’s FIRST SECOND inquiry where (1 unit space = 1 unit (time^2) )
WILL validate the Merton College Calculators ‘MEAN SPEED THEOREM’ (circa1350) using Sir Isaac Newton’s (1643-1727) Universal Law of gravity found some 300 years after College Calculators.

We begin with the square of Pythagoras,  (a^2) + (a^2) = (((diagonal))^2); 
diagonal = (Sqrt[2(a^2)]); =a(Sqrt[2])

Any perfect square diagonal has meter in direct proportion with its side as (a(Sqrt[2])).

Constructing a CSDA Space and Time square for Uniform Acceleration

Let central force F be unit circle center. Let ((ℼ/2) spin axis) = the meter of time, and (ℼ, 2ℼ plane of rotation) = the meter of of space. Let free fall vector ((OverVector[Fd])) be a snap shot 1st second uniform acceleration experience (from 0 momentum to some accelerated velocity v). Free fall displacement space = (s). Let the unit circle be the surface acceleration curve of M1.

Treating kinematics parts, uniform acceleration, time, and resultant meter of free fall space using a compound central force free fall vector (Fd).

To use curves to determine how far, we convert (u) into a radius and scribe a displacement curve above surface acceleration curve of central force (F) passing through the tail (d) of free fall vector (u). 
We now have two plane geometry descriptive terms: Radius  Fd= (Sqrt[2]) ; and Curvature  @ d = (1/(Sqrt[2]))a number.

(32ft/sec×((((1/(Sqrt[2]))))^2))=16 feet 1st second experiance.

(9.8m/sec×((((1/(Sqrt[2]))))^2))=4.9 meter1st second experiance.

We can now assign specific space-time g-field acceleration curvature 
to Sir Isaac Newton’s Central Force point mass (F) producing the surface acceleration curve of (Subscript[M, 1]).

((1st second free fall experience× ((((((1/(Sqrt[2]))))^2)))^(-1)) =  acceleration curvature of any point  mass (F))

SLIDE 8 : Using Acceleration Curvature and High School falling Physics.

GIVEN:     acceleration curvature of planet earth point mass is ((1/16)).

PROBLEM:	Find impact velocity and time to impact for an object, released via free fall, 400 feet above the earth’ s surface.

SOLUTION: The displaced position curve with respect to surface curve of gravity field potential is presented as 1/400.

TIME OF FREE FALL
Solve[(1/16)==(1/400)*(t^2),t]
{{t->-5},{t->5}}
We see time of free fall is five seconds. To find terminal velocity, double the radius of position displacement curve and divide by time

TERMINAL VELOCITY
((400*2)/5)=160ft/sec

MEAN SPEED THEOREM

TERMINAL VELOCITY (5th time frame/earth)
((400*2)/5)=160ft/sec

For 5th time frame (Uniform Acceleration planet Earth) fall velocity will be 160ft/Sec.
If we divide free fall accelerated velocity (160 ft/Sec) by 1/2; essentially an average where (initial v = 0), we acquire a fixed velocity of 80ft/Sec. for replacement value of initial free fall momentum 0ft/Sec, showing us ‘Mean Speed Theorem’ equivalence.  If at initial parameter we entered space time (Earth) with fixed velocity of 80 ft/Sec over the 5 second period we will move 400 ft, the same as uniform acceleration imparted to a stand still comparative accelerated with free fall potential for 5 seconds to 160 ft/sec reached at the 400 ft. mark.

END PART 1 UNIFORM ACCELERATION:

DEMONSTRATING UNIT CIRCLE CAPACITY TO ACQUIRE PHYSICAL PROPERTIES OF A SPACE TIME WAVE FORM (USING SOUND). 
GIVEN: [1st second history of a sound curve; space time earth:1100 ft and 1sec]
PROBLEM: use parametric geometry construct wave form displacement space for 1/2 second and 1 second from source/origin.

METHODS: We begin with a CSDA construction where unit circle is first second history of a sound curve from a central source. We next construct CSDA unit parabola time/energy curve and find Cartesian location on time curve for ½ second and one second wave displacement from source. 
The initial first second dependent vertex radius of curvature of the CSDA focal radius is a constant (2p), making the 1st second dependent time/energy sound wave vertex curvature: ((1/(2p))). This will be my reference base to map wave front displacement from a central source. 
Construct abcissa event for leading edge wave form at (1/2)and 1 second.

(the yellow curve is the system osculating circle providing curvature value for initial CSDA time energy curve).

{2Cos[t]+0,2Sin[t]-1} -> time energy curve osculating circle for radius of curvature

ParametricPlot[{{Cos[t],Sin[t]},{2Cos[t]+0,2Sin[t]-1},{t,((t^2)/(-4))+1},{.5,t},{1.0,t}},{t,-π,2π},PlotRange->{{-1,(7/2)},{((-5)/2),2}}]

CSDA analytics for wave form mechanics of sound.

as focal radii move on the time curve they become leading edge abscissa of wave form propogation. To find Cartesian definition of position following focal radii on the time curve we use analytic/parametric geometry.

we need energy tangent slope to track wave motion through space using CSDA dependent time curves: (1st derivative of dependent curve for energy tangent slope will be: ((-t)/2)

(∂_t)(((t^2)/(-4))+1) ⟶ -(t/2)

tracking sound wave motive energy

Set up a template for dependent curve and set in (time value (1/2) s) to find  ordinate value for focal radius position on the time curve. We have Cartesian coordinate pair ((1/2)for abscissa time, and 15/16 locator)

((t^2)/(-4))+1/.t->(1/2) -> ((1/2), (15/16))

Use analytic geometry point slope eq to find energy tangent. Slope term is first derivative of dependent curve (((-t)/2)). Set up analytic point slope eq for a line using first deriv term for slope,

Solve[y-(15/16)==((-(1/2))/2)(x-(1/2)),y] -> {t,(1/16)(17-4t)} -> post energy tangent into the construction

Same for time analytics of second 1. Cartesian pair (1 s time,  (3/4) locator  ), use pt slope eq to find energy tangent and post to const.

((t^2)/(-4))+1/.t->1 -> (1, (3/4))

Solve[y-(3/4)==((-1)/2)(x-1),y] -> {t, (1/4) (5-2 t)} -> post energy tangent into construction

ParametricPlot[{{Cos[t],Sin[t]},{t,((t^2)/(-4))+1},{.5,t},{1.0,t},{t,(1/16)(17-4t)},{t,(1/4) (5-2 t)}},{t,-π,2π},PlotRange->{{-1,3},{-1,2}}]

using tangent slope to find leading edge wave form displacement.

Let the initial focal radius (p) have measure of 1100 ft. Every parabola vertex radius of curvature is (2p) making dependent parabola vertex curvature = ((1/(2p))). Set up a direct proportion using the initial dependent motive energy curvature of space as 1st and 2nd proportional.  The 3rd proportional is the variable time (t) and the 4th proportional is the first derivative denominator. Solve for (t) to find wave front displacement from central  source.

(1/(2 ))second wave front position from source

Solve[(1/2200)==(((-1t)/2)/2),t]

Convert curvature term to abs radii value to find leading edge wave form displacement from a central source.

(((-(1/550)))^(-1)) -> Abs[-550] = (550 ft in (1/2) sec)

1 second wave front position from source

Solve[(1/2200)==((-1t)/2),t]

Convert curvature term to abs radii value to find leading edge wave form displacement from a central source.

{{t->-(1/1100)}} -> (((((-1)/1100)))^(-1)) -> Abs[-1100] -> (1100 ft in 1 sec)

PART 2
CONSTRUCTING CHANGING ACCELRATION 
(FIELD CURVES) OF M2 ORBIT

A quick demonstration constructing a standard model gravity field space time square for our earth/moon system.

Defining CSDA (Curved Space Division Assembly) via RULES of CONSTRUCTION

FIELD POTENTIAL IS CONFINED TO UNIT CIRCLE

UNIT CIRCLE IS SYSTEM INDEPENDENT ENERGY CURVE

FIELD MOTIVE ENERGY IS TRACKED ON DEPENDENT UNIT PARABOLA TIME CURVE

This time square also has two diagonals. The linear diagonal will be system dependent curve latus rectum connecting sides of the space time square between orbit limits, and the curved time diagonal connects corner limits of orbit system space time energy.

CHANGING ACCELRATION (STABLE (Subscript[M, 2]) ORBIT).
SPACE AND TIME SQUARE

This is a CSDA construction of orbit gravity in action: Abscissa sides = position limits of event radii and the energy level of position limits = ordinate sides providing g-field orbit (changing acceleration) space/time square. 
 Limits of high energy curve is red, low energy curve limits are blue.

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

Figure 2: parametric space time square (orbit parameters slide 13, page 12).

Figure 3:  g-field spin and accretion (CSDA) space time sq. and energy tangent.

Another view of space time square centered on positive side of the LR diameter with curved space event slope (-1) energy tangent tracking g-field spin and accretion. Note changing acceleration space time sq is removed from and does not touch M1 potential curve.

STEP 1: BUILDING EARTH AND MOON SPACE AND TIME SQUARE. NASA data:

Orbital parameters (for lunar orbit about the Earth)

Semi major axis        384400 km 
       
Perigee        	         363300 km 

Apogee 	         405500 km

orbit KE in km/Sec        
Mean orbital velocity (km/s)        1.022 km/s

To construct a standard model CSDA, I convert NASA square space (position) data into relative unity data for curves. How do we do this?

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)(high e level), 20521, \[Placeholder], 0.1068}, {f(a)(low e level), -21680, \[Placeholder], -0.1128}, {focalradius(high e), 363879, 2r-f(r), 1.8932}, {focalradius (low e), 406080, 2r-f(r), 2.1128}})

To convert NASA parameters (green column) into comparative unity ratios (orange column) of a CSDA curved space reference frame use the average radius as denominator of all conversion comparatives. Answers for CSDA curved space converted units (yellow column) will be returned when event parameters are set as numerator. To standardize CSDA g-field comparatives concerning fixed potential; multiply the independent potential curve by 1 and the dependent orbit period parameter curves by two.

CURVED SPACE ORBIT PAIRING (position, position energy). Using the dependent curve focal radius we can follow position and position energy of orbit.
CDF Code by Abraham Gadalla

ITEM 1: LATUS RECTUM RESIDENT CURVE

changing energy tangent slope 
CDF Code by Abraham Gadalla

finding energy curve rotation momentum using energy tangents (HIGH ENERGY) focal radius = 1.8932, event radius = 1.8902
(NASA) data for high energy limit = 1.076 KM/SEC

Solve[(1/1.022)==(((((1.8932/2)))^(-1))/v),v]

Figure 3:  g-field spin and accretion (CSDA) space time sq. and energy tangent.

TO CONSTRUCT A CSDA; DIVIDE THE GRAVITY FIELD IN HALF AROUND THE CENTRAL FORCE ((Subscript[M, 1]))
 USING TWO CURVES.

INDEPENDENT HALF (OUR UNIT CIRCLE)

(1Cos[t], 1Sin[t]): A closed Constant  Acceleration surface curve used to study g-field potential and mechanical meter of “weight”(ie; surface acceleration phenomena of our earth).

DEPENDENT HALF (SBG OPEN CURVE UNIT PARABOLA)

t, ((t^2)/(-4(p)))+r) Used to meter Changing Acceleration phenomena; altered kinetic energy that curves orbit motion.

HOW DO WE CONVERT EUCLIDEAN SQUARE SPACE CURVES INTO EUCLIDEAN CURVED SPACE PHENOMENA?

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

Next add some Gfield mass/volume ratio making FIELD POTENTIAL summed content of the unit circle. Adding mass/volume ratio as content to our unit circle changes first degree linear tangents into second degree curved influence of gravity field potential.

BASIC GRAVITY FIELD CURVED SPACE DIVISION ASSEMBLY

The tangents now curved; become a function of the mass volume content with vertex assignment of the system energy curves at the poles of our unit circle; using spin and rotation accretion symmetry we can now map a time and energy g-field motion curve.

INDEPENDENT CURVE⟶1Cos[t], 1Sin[t]  
 DEPENDENT CURVE⟶ {t,((t^2)/(-4(p)))+r}

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

to 21st century perception of gfield spin as (range) and accretion rotation as (domain) using a Computer Algebra System. The curved space directrix is not a limit on field range but controls the reach of gfield potential.

Basic CSDA; spin and rotation map of (gravity field)

CARTESIAN ORBIT PAIRING (position, position energy).
CDF Code by Abraham Gadalla

ITEM 1: LATUS RECTUM RESIDENT CURVE

item 2: changing energy tangent slope 
CDF Code by Abraham Gadalla

finding energy curve rotation momentum using energy tangents (HIGH ENERGY) focal radius = 1.8932
(NASA) data for high energy limit = 1.076 KM/SEC

Solve[(1/1.022)==(((((1.8932/2)))^(-1))/v),v]

finding momentum using (LOW ENERGY TANGENT) (NASA)

MIN ORBIT momentum = 0.964 km/sec (NASA)    FOCAL RADIUS MAG = 2.1128

Solve[(1/1.022)==(((((2.1128/2)))^(-1))/v),v]

.967-.964 → 0.003 (diffrence of 3 meters/sec FASTER).

Part 2: 
CONSTRUCTING SPACE AND TIME

Let’s construct HIGH and LOW mechanical energy curves for our earth moon system.

BUILDING EARTH AND MOON SPACE AND TIME SQUARE
I VISITED NASA WEB SITE FOR Moon/Earth Parameters. WE NEED HIGH AND LOW ENERGY PARAMETERS AS WELL AS MOMENTUM FOR THE AVERAGE ENERGY CURVE.

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

(363300/384400)*2.0

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)(high e level), 20521, \[Placeholder], 0.1068}, {f(a)(low e level), -21680, \[Placeholder], -0.1128}, {focalradius(high e), 363879, 2r-f(r), 1.8932}, {focalradius (low e), 406080, 2r-f(r), 2.1128}})

abscissa sides of square: 2.1098-1.8902 = .2196

ordinate sides of square: .1068-(-.1128) =.2196

FINDING ENERGY LEVEL f (r): ((t^2)/(-4(p)))+r/.t->event radius
to find energy level; use a CAS dep crv template and plug in event r

FINDING FOCAL MAG (2p = Radius of Curvature at Time Curve Vertex)
AND diameter of independent field potential curve = 2 units;
∴ focal radius mag = 2units - f (r)

Going from orbit radius of curvature to 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]) (earth)and (Subscript[M, 2]) (moon)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

(((((((1/2.1098)))^2)*4))^(-1))

PART 3: CONSTRUCTING AND SHAPING INITIAL MOTIVE CURVE PROPERTIES OF GIVEN FIELD POTENTIAL

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 congruent shape with 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 negative slope event (- 1) on the gfield time curve. This cooperative endeavor gives us a two unit event radius happening at (-1) slope event time and energy curve.

Notice the inverse connection: G-field M1 unity curve contains the complete history of micro infinity event curvature; and outside the surface curve of M1 potential we find the complete history of macro infinity square space event radii. Plane Geometry Inversed connections are the g-field tether connecting M2 event radius motion in square space with the micro infinity curving phenomena of gravity field M1 potential.

Prove shape of average motive curve = shape of potential curve.
1. Construct the range of gfield potential as a tangent limit through orbit space. (t, 1)
           2. Construct shape of potential curve (curvature = 1) about central force (Subscript[M, 1]); 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 using focal properties:  (focal radius - potential radius) ⟶  2 - 1 = 1 unit

b) Radius of motive curve using inverse square law with respect to the energy diameter on the latus rectum:
(event radius = 2) -> [(((((((1/2)))^2)×4(1)))^(-1))⟶1 unit]

QED; (curvature of potential curve = 1); and (curvature of average 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 shaping radius = .8932

Sir Isaac Newton's Inverse Sq Law: event radius = 1.8902
(((((((1/1.8902)))^2)*4))^(-1))⟶ KE shaping radius will be 0.893214

{post high energy parameters {0.8932 Cos[t] +1.8902, 0.8932 Sin[t] + 0.1068} on to gfield time curve for high energy motive curve.

CONSTRUCT LOW ENERGY MOTIVE CURVE OF EARTH MOON

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

Sir Isaac (event radius=2.1098):(((((((1/2.1098)))^2)*4))^(-1))=1.1128= shaping radius low energy curve

Low energy parameters[1.1128Cos[t]+2.1098,1.1128Sin[t]-.1128}...post construction on to gfield time curve and we have the low energy motive curve

Step 7: CONSTRUCT BOTH HIGH ENERGY FIELD TANGENTS OF EARTH/MOON SYSTEM

({{high energy event radius, =, 1.8902}}) AND          event energy = 0.1068

energy tangent use point slope equation where slope is first derivative of dependent curve: Solve [y-(.1068)==(1.8902/(-2))(x-1.8902),y]⟶ {t, 0.1068-0.9451 (-1.8902+t)}

insulator tangents separate opposing field forces of attraction and escape; are hypotenuse related right triangles:

insulator tangent: Solve[(1/y)==(1.8932/(.1068)),y]⟶ordinate;.0564      ; Solve[(1/x)==(1.8932/1.8902),x]⟶ abscissa; .9984

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

Step 11: Construct Both Low Energy Orbit Tangents and Motive Curve shaping the orbit of (Mars). 
 STANDARD MODEL DEMONSTRATION FOR G-FIELD ANALYTICS USING A CSDA UTILITY

event radius = 2.1870         AND event energy = (-0.1957)               AND focal radius = 2.1957

({{CSDA PARAMETERS sun/mars}, {1.8131}, {2.1870}, {2}, {1}})= (((((((1/2.1870)))^2)*4))^(-1)) =  1.19574  ({{CSDA parameters earth/moon}, {1.8902}, {2.1098}, {2}, {1}})

Solve[(1/x)==(2.1957/2.1870),x]=0.9960

{{x->0.9960}}

Solve[(1/y)==(2.1957/((-0.1957))),y]=-0.0891

{{y->-0.0891}}

Solve[y-(-0.0891)==((((0.1957/2.1870)))^(-1))(x-0.9960),y]⟶ -0.0891+2.3365 (-2.187+x)⟶
{t,  -0.0891+2.3365 (-2.187+t)}

CONVERGENCE POINT OF MOTION ENERGY: 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.

PROOF ALL TANGENT INTERCEPTS = EVENT RADIUS/2 ON THE CURVED SPACE DIRECTRIX

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]

{t->0.9451}

(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

THANKS;    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):

alexander@sandboxgeometry.com

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

ParametricPlot[{{Cos[t],Sin[t]},{t,((t^2)/(-4))+1},{2.10978,t},{t,(-0.112791)},{t,-0.1128-1.0549 (-2.1098+t)}},{t,-π,2π},PlotRange->{{(3/2),(9/4)},{((-1)/2),(1/2)}}]

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

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

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 FOCAL RADIUS SLOPE

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 SLOPE

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]

{t->0.9451}

(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

(((((((1/1.8902)))^2)*4))^(-1))

SirIsaac:(((((((1/2.1098)))^2)×4))^(-1))=1.1128

(((((((1/2.1098)))^2)*4))^(-1))

poincaire (n-body demonstration inner group orbit limits with respect to sun).

first 4 solar space curves; Mercury (black), Venus (green), Earth (blue), and Mars (red).

ParametricPlot[{{206560000,t},{249120000,t},{227840000,t},{t,(t^2)/(-4(113920000))+113920000},{t,20286236},{t,-22273764},{107370000,t},{108850000,t},{46040000,t},{54055000,t},{69860000,t},{147070000,t},{152070000,t},{74785000,t}},{t,-25000000,250000000}]

Mars orbit limits marks the field boundary of the inner group potential. ASI is a collective curve of gravity field potential. Jupiter and it’s ASI lies outside period limits of Mars, this effectively sets field potential curve controlling outer group separate the inner grouping. This makes potential curve controlling the orbit of the 5 th solar curve (Jupiter) inclusive of all inner group potential phenomena.

OUTER GROUP POTENTIAL AND MOTION DISTRIBUTION

Yellow highlight will be Pluto and Mars parameters
(Jupiter; green). (Saturn; orange). (Uranus; fuscia). (Neptune; blue).

ParametricPlot[{{4443000000, t}, 
   {7375000000, t}, {5909000000, t}, 
   {t, t^2/(-4*2954500000) + 
     2954500000}, {t, 1284145541}, 
   {t, -1647854459}, {206560000, t}, 
   {249120000, t},{389070000,t},{740480000, t}, 
   {815800000, t},{713500000,t}, {1349500000, t}, 
   {1504500000, t}, {1435150000,t},{2738300000, t}, 
   {3002300000, t},{2249950000,t}, {4463000000, t}, 
   {4536800000, t}}, 
  {t, -1700000000, 7400000000}]

High energy orbit limit of Pluto precedes high energy limit of Neptune and limit linear demarcation is exaggerated because at this numerical description both orbit limits deceptively coincide.

OUTER GROUP PARAMETERS:

JUPITER (green)

1.  {740480000,t}->(π; high energy limit)
	2.  {815800000,t}->(α; low energy limit)
	3.  {778140000,t}->r-aorb...(time curve latus rectum energy diameter)
	4.  ASI of 5th solar space curve; Jupiter...[ 778140000/2 = 389070000]

SATURN (orange)

1. {1349500000,t}->  (π; high energy limit)
	2.  {1504500000,t}-> (α; low energy limit)
	3.  {1427000000,t}->r-aorb...(time curve latus rectum energy diameter)
	4.  ASI 6th solar space curve; Saturn...[1427000000/2 = 713500000]

URANUS (fuscia)

1. {2738300000, t}→  (π; high energy limit)
	2. {3002300000, t}→  (α; low energy limit)
	3. {2870300000, t}→ r - aorb...(time curve latus rectum energy diameter)
	4. ASI 7 th solar space curve; Uranus...[2870300000/2 = 1435150000

NEPTUNE (blue)

1.  {4463000000,t}->(π; high energy limit)
	2.  {4536800000,t}-> (α; low energy limit)
	3.  {4499900000,t}->r-aorb...(time curve latus rectum energy diameter)
	4.   8th solar space curve; Neptune...[4499900000/2 = 2249950000]

PLUTO
	1.  {4443000000,t}->(π; high energy limit)
	2.  {7375000000,t}->(α; low energy limit)
	3.  {5909000000,t}->r-aorb...(time curve latus rectum energy diameter)
	4.   ASI 9th solar space curve; Pluto...[5909000000/2 = 2954500000]