Electronic Sensor Prototype to Detect and Measure Curvature Through Their Curvature Energy

: Through of the concept of curvature energy, and the curvature theory on homogeneous spaces is designed and constructed an electronic sensor prototype to measure curvature in 2, and 3-dimensional spaces using the programming of shape operators in micro-controllers and the value of their energy integrals along the curves and geodesics in their principal directions. The curvature obtained under the sensor device is their spectral curvature given in voltage in 2-and 3-dimensions and their perception of curvature through electronic signals on a curved surface is the curvature energy that is detected in the displacement of an accelerometer on the curved surface.


Introduction
In the study of the field observables are of major importance from a point of view of their geometry, the curvature and torsion of a space affected for a field 1 , which are the geometrical invariant that defines their shape and orientation.
In our design analysis, we will consider like model of curvature, the obtained one by the Gaussian curvature, which involves from the mathematical point of view, curvatures along geodesic, curves on surfaces and space, sections of curved surfaces and bodies in the space, considering the design of an operator of shape based at the beginning in the geometry of the sphere 2 S . Likewise, the value of certain integrals on cycles (invariant classes of the space 2 ) and 1For example the gravitational field that produces a geometrical scenery in the space-time affecting to this by the only presence of the matter. Likewise, their curvature is the field observable deduced by the curvature tensor of the corresponding field equation of Einstein.
2 For example, circles to the − 2 sphere, horocycles to a hyperbolic disc, planes to Euclidean space, etc., where can be evaluate the integrals of lines, surfaces, or considering the case of − 2 surfaces, on which is measured the curvature through the two principal directions representing these two values, as the maximum and minimal value of curvature of the surface determining their shape [1], can be established the curvature interval , 2 1 k k k ≤ ≤ [1], on which we design the programming of our sensor. This obeys to a question of design of the sensor, and also of the space perception of the controller on the accelerometer device, which must to use a modulation space with a domineering energy condition given by [1][2][3][4][5][6][7]: where V, is the applied potential energy of curvature 3  is the area of the surface and , h their mean curvature and the last integral correspond to the curvature energy employed to measure the roundness in their displacement to along of two principal directions.
Through certain studies of the models of Gaussian and normal curvatures to determine by Hilbert inequality their curvature energy, we can on certain bound of roundness obtained through of the implementation of the spherical operator (spherizer [7,8] use the idea of the Radon transform to determine through cycles and their co-cycles the curvature of the surface or body having the Hessian of the − 2 fundamental form , Ω of the shape operator as a fundamental part of a censorship when we consider functions that are − 2 differentiable functions in the corresponding principal directions. The development of research in theory of curvature in homogeneous spaces [8], has established that the measure of curvature can be obtained as an extrinsic curvature of the space classes (cycles) which have a curvature measure well defined. From a point of view of the signals and system analysis the utilization of these cycles could be translated in the context of the Fourier analysis in the application of energy pulses ), , ( y x π [7] that can reproduce in the infinitum the measure the curvature through of their energy spectrum (see the figure 1). Fig. 1. The shape of a2-surface (hyperbolic paraboloid or "ride chair") is determined through energy Gaussian pulses [9]. Remember that the curvature is the more important geometrical invariant that determines the shape of a space. 4 Integral operator that involves a spherical map in the sphere . 2

S
The spherizer will be very useful in our design of sensor device as superior bound of the factor of gravity considered in the design of our accelerometer. Likewise, These cycles must be invariant under translations and  rotations 5 , to use them as symmetrical patrons in electronics and photonics of the energy signals that we want consignee in curvature information. Likewise, considering the curvature as a fundamental − 2 form whose representation in a Hilbert space (energy space) is given by , Bulnes) [8][9][10]) we have that: which is an "energy" representation of curvature. This permit us generalize the idea of the curvature as field observable to a level of their energy, having the concept of curvature energy, that is to say, the domineering energy in the action of the field on a curved space [11] to displace a particle on said space (see the figure 2).    Of fact, this idea is consigned, to the case of solid state electronics, in the first curvature sensor prototypes published in [7], with the application of the censorship given by (1) and followed after in the research of field curvature and torsion to quantum gravity [11][12][13][14] to the prospective of a sensor of quantum gravity. Also with this idea were designed other devices of photonics and opto-electronics nature considering light waves in the detection and measurement of curvature [15][16][17]. In this sense, is published a result with applications in opto-electronics and photonics in [17]: Theorem (F. Bulnes) 1. 1. [17,18]. The Radon transform of the Gaussian curvature whose detection condition is the inequality (censorship 7 ): and using the signals the curvature measured by light beam, is: 7 Theoretical sensor of curvature in presence of the incurve and detected by a wave of light [17,18].
Proof. [17,18]. A design and construction of a sensor device needs the recognition on the part of the accelerometer of the property of roundness of a surface or body (see the figure 3), their perception of this roundness from the point of view of the signals of the sensor that the accelerometer will involve in their advance (displacement), and their aptitude to cause information in real-time, according to this perception as curvature of the surface or body (reading of the sensor). Then we want the following theorem [7]: with the Gaussian pulse. Then an energy co-cycle (in the Radon space 8 ) is that, whose energy area is: Proof. We consider the projection of the 2-dimensional Gaussian pulse ), , ( to a maximum circle , ε C of the sphere of radius 1 = ε , [7] which bounds the length of arc of the Gaussian pulse in the measurement of curvature in the surface in one direction (see the figure 3).
Then considering the voltages of output of our accelerometer device as the produced by the displacements when is considered the length of perception in a Gaussian pulse in the space (given for the length p ) we can to define the sensoring advance of the accelerometer on surface (see the figure 4).

Development and Construction of the
Electronic Sensor Prototype of Curvature. Optimization of the Curvature Sensor

Design and Fabrication of the Electronic Board of Micro-controller Pic16F877A Obeying the Gauging of Inter-phases with Accelerometer Actions
As has been demonstrated in the theorem page 120, [7], the fabrication of our microcontroller must to be designed and gauged accord to the accelerometer actions, that is to say, the detected curvature energy data (given in voltages) must to be interpreted by the microcontroller in outputs of principal curvatures considering the two principal directions of the tangent space on surface that varies. The axis , Z must to be interpreted as − g cell factor in the monitoring action through cycles of energy (pulses) obtained when is varied the height in the displacement of our accelerometer (see the figure 5). Although also, in this − Z direction the output voltage in the sense strictly of the micro-controller is a principal curvature if we consider the theorem established in [7]. This design gives opportunity of to create an air version of this curvature sensor where can be used the Z-direction voltage output (see the figure 6). This variation of height can be considered as variation of the normal vector to the tangent plane, establishing through of the direction variation speed their normal curvature where , u is an unitary tangent vector to a surface M (see the figure   6). ) (u S p , is the shape operator depending the unitary normal field ), ( p U in a neighborhood of . M p ∈ This will be of importance to define cycles of certain "length" inside of this neighborhood to the displacement perception of the accelerometer which will must to consignee spatial variations in curvature energy, this last in the micro-controller.
To this goal, were realized settings in the recorded programming in the micro-controller to the sensitivity of our accelerometer to that their displacement could be calibrated in the neighborhood established by the shape operator ) (u S p , and with it could be implemented the accelerometer sensored reaction our, with the displacement to can realize the measurements while our device is in movement. These settings consist in an electronic plate or board of micro-controller fabricated in laboratory considering a ceramic board of Siliceous with nodes aligned to constant distribution of charge annulling the decreasing of potential to Through Their Curvature Energy electronic elements [2][3][4]7].

Fig. 6.
Unitary normal field is reproduced by the height variation in the surface relieve by the air version of the curvature sensor, when the accelerometer is installed in a drone. In this case, we have that in a convex surface we have We need to design and construct an optimal electronic board in the accelerometer, whose voltage output data must be more exact and that responses with the displacement, not only when is in without movement (to future researches we want to implement an autonomous movement system of the accelerometer with certain velocity). To these goals in short and long terms, was modified the printed circuit to the electronic board, obtaining measurements in positive and negative voltages depending of the orientation and doing possible the exact measurements during their movement (see the figure 7) [15, 20,21].

Displacement Perception Condition
We have obtained a displacement perception condition of curvature given by the "curvature length" [7] (see the and establish that the superior extreme given in the inequality (11) is a condition of curvature energy of majored curvature given on a geodesic , γ of our cycle through their length 9 (arc

E T z
That is to say there is "obstacle" in this surface that generates deviations. Shrinking the loop by sending 0 , → t s (to two limits as (13)) gives the infinitesimal description of the following deviation, which finally define curvature [8]:   θ we have by definition [1,8] ))), Here , κ is curvature length implemented by (10) [7]. In the next section the sensor field , η (which has three enters as vector field )), will be characterized by the transference function ), (t h calculated through capacitance voltages of the electronic characteristics of sensor in the three directions of the accelerometer. Then we can give the corresponding programming electronic design of the complete system of the microcontroller using the Master Prog recorder which was written in the Ram memory of the microcontroller PIC16F877A (see the figure 10).
which is an isomorphism. Then the variation of the length is given for ).

Experiments
The data will be observed and registered through of The first step is to observe the values through a flat surface are limiting to a 5V conversion as maximum, to gauging effects of our sensing and their dependency of displacement (see the figure 11 A)). This produce us a sensitive platform in the sensor to obtaining of exact measures starting in a flat space which without curvature must to give 0 in their screen led (see the figure 11 B)). However, and due the physical conditions the sensor start with little deviation of order 0.0000025 (sensitivity at typical conditions divided for 1.5g-cell of the accelerometer at typical conditions).
We can to view the voltages in the three axis in different position in our LCD screen (see the figure 12).
The statistical data are considered to a short trajectory as showed in the figure     During the displacement of the accelerometer in an amorphous surface is can to visualize in the led of the LCD, that the measures are realized of the speed and continuous form, we realizing of that the modifications in the configuration of the accelerometer MMA7361, are correct and efficient, thanks to the use of Gaussian pulses of high efficiency as has been demonstrated in the propositions 1. 1, and 2. 1.

Spectral Curvature
The detection and measurement of the curvature is realized using the value of the integrals of a field interacting on the geometric pattern along their surface doing it on signals of finite energy defined in the proposition 1. 1, and that they will code the information of curvature in a spectral space ))), [20]; through of the signals given in the frequency established in our detector device that detects and measures curvature.
If , s X is the electromagnetic field of EDMC/CURVE device (sensor field 10 ) then we use the Radon transform such and as was mentioned in the theorem 1. 1, with the corresponding Hessian to calculate the curvature of a region , S of . M But this region is the energy region of the rays of light used to determine the measure of curvature with constant electromagnetic fields. This agrees with the Gaussian curvature, which use an identification of the sphere of dimension four [17] and solutions through of the use of the operator , MAX L and that can be obtained by the formula [7]: is the corresponding Maxwell field required to the bundle of light of the connection D . To our very particular case (dimension two) our spectral Gaussian curvature will be: Considering the tempered distribution [22,23] ), , (

Transitory Analysis of Response and
Border Condition to C , and C V , to the

Co-Cycles of Curvature Energy
We establish a linear model of response in voltage outputs of the sensor displacement as was explained in the sub-section 2. 2, which obeys to a differential equation of − RC type. We consider the following Kirchhoff law: having that for each circuit element of the − RC circuit with variable capacitor, the equivalent voltage is null: The solution to the differential equation given by (19)  , that is to say, the space is understanding through Gaussian pulses. Enters of the Hessian matrix must be curvature spectra of these pulses (see the figure 15). We consider the differential equation to inputs and outputs system: where we have the transference function: where , is given by the system:    Our curvature sensor operates in a range of voltage until , 5V permitting exact detections of curvature by the accelerometer with outputs given by ).
The negative part must be interpreted to space zones whose curvature is negative.

Units
We consider now, the diverse aspects of dimensionality of operating and physical operating conditions (see the table 2), as also approximations through non-harmonic analysis to the study the energy cycles using cylindrical functions in their re-construction. This last analysis helps to understand the interacting of physical space of the system and their cycles approximating the space, in this case the shape of the space. ∫ In this step we have that the phase of our Gaussian pulse in the spectral space is a Bessel function, that to our sensor system we have The physical properties of operating of our sensor function with the phase defined to our voltage C V , given by Bessel functions that determine the variable modus of operating of the capacitor can be seen in the Table II. This phase involves the voltage

Advanced Studies in Curvature as Field Observable: 2-Dimensional Curvature Model through Hyperbolic Waves
An application to aero-spatial technology presented and published in the year 2002-2003 [18,19], using the spectral model of curvature given in (17) according with the physical concept of curvature energy [7] to a 2-dimensional representation of curvature as tempered distribution of the space-time, using the 2-dimensional hyperbolic model of the space-time as hyperbolic disc and cycles given by hyperbolic waves is the spectral curvature:  Fig. 17. Line spectra of curvature to the 2-dimensional hyperbolic model of space-time. Observe the long infra-red spectra according with the conclusions of the expansion of the Universe.

Invisibility of Objects
In certain transitory process of the light refraction effects produced for certain materials and their optics mechanisms the curvature can be fundamental to produce invisibility effects in an object, where certain curvature energy spilled in an object is diffracted, producing an effect of invisibility apparent, which is a deviation of the light that permits to observe the objects in their colors and details.
Of a more precise manage we can affirm that the effect is produced due to that the light field can be curved through of certain curvature energy that have the properties of curve the light, at least for the presence of some of the atoms of the certain materials that compose the object, producing an effect of curvature in the light, which generate an apparent invisibility (similar to like in the eclipses, although by distinct reasons, since in this last is due to the gravity).

Applications to Fine Movements and Control of Drones
The movement of drones to special mission are required with major precision and fine flight attitude of the drone in remote control can be optimized with the help of the curvature sensor which can establish in the drone a change of patterns of flight interpreting through curvature as smoothed movements and the object approximation as obstacle to the field sensor doing to vary the flux of their energy in their accelerometer.
A concrete application is to control a cup drone designed as helicopter to realize work of vigilance in a building in construction (see the figure 19).

Conclusions
The electronic prototype of solid state presented in this research has been developed through the curvature energy considering that in physical level, the perception of the curvature in the space has that to be given in terms of an electronic characteristic that has a relation with the geometrical enthrone where the direction change speed of the space due to the existence of curvature is established and perceived through of change of sensor field and their flux on the surface where is realized the measure. Considering that the Gaussian curvature can be approximated through the electronic pulses, these as special functions which are invariant under transformations of the space and their electrical characteristics, is constructed the curvature energy version of the Gaussian curvature and the mean curvatures using the principal curvatures as tempered functions whose spectra is an element of the space ))), where the transference function implicit to the system, ), (s H establish the relation of displacement with the system outputs in function of a − RC characterizing on the circuits where variable regime of capacitance will produce the sensor effect wanted. Then the voltage due the capacitance is , These aspects can be majored under the use of certain semi-conductor materials that can to produce in the circuit of micro-controller low impedance to requirement of control in resonance by frequencies, for example in the accelerometer [5,21]. Also the requirement of current to the feeding of Micro-controller.
This prototype of curvature sensor is the first of many others that will be obtained under the same philosophy and principles on curvature energy where we want to establish a sensoring of the space through their energy, considering that the space and time are unified in an energy enveloping in all levels of measurement through their pure energy. Sub-sequence prototypes will be realized under electronics technologies versions with the goal of measure curvature and torsion as field observable and not as simple characteristics of the space. This last and after of many developments of sensor prototypes and as finally prototype will be designed under the same principle, curvature energy; a curvature and torsion sensor that has the work of measure the field interactions as particles that deform the space and time and to obtain a curvature sensor to detect and measure quantum gravity. To this case could be developed a SOIC-type circuit integrated in the micro-controller.
In the way, the applications will be much and diverse, for example, applications in MEMS developments and nano-materials, non-symmetrical fields technology, particles interferometers, special mimetic and optical effects devices, optimal control of drones, distributed generation considering the curvature to the design of special surfaces and tunnels, etc; and new metrologies.
Also we could to be obtained, under this same philosophy, applications in medicine, since curvature energy could establish deviations of energy, puddles or jumps which can be re-interpreted under spectral curvature as sickness in the body. To it, we will need curvature sensors that measure change of energy under field observables. Through Their Curvature Energy Law B, Principal of TESCHA, Evaristo Vázquez, Hernández, LC, B, Financial Sub-principal, and Rene, Electronic Eng., Electronic Division Chief, by their material support and facilities to realize this research.

Technical Notation
− K Curvature as general concept of roundness property. Also used in the paper as Gaussian curvature in a point . Area.
-Space of spectral transformations on curvature forms given in space Space whose curvature is measured. In our study represent dimensional surfaces or dimensional bodies.
Dimensional sphere. Also is the dimensional sphere used in the spherical map to design our curvature sensor -Evaluation of curvature radius from the product from their inverse principal curvatures.
Curvature form in the and dimensional spaces  Program designed to create electronic circuits to the after the design print circuits to one or two faces.