Normal Vector Of Plane Research Articles

Rational minimal-twist motions on curves with rotation-minimizing Euler–Rodrigues frames

A minimal twist frame ( f 1 ( ξ ) , f 2 ( ξ ) , f 3 ( ξ ) ) on a polynomial space curve r ( ξ ) , ξ ∈ [ 0 , 1 ] is an orthonormal frame, where f 1 ( ξ ) is the tangent and the normal-plane vectors f 2 ( ξ ) , f 3 ( ξ ) have the least variation between given initial and final instances f 2 ( 0 ) , f 3 ( 0 ) and f 2 ( 1 ) , f 3 ( 1 ) . Namely, if ω = ω 1 f 1 + ω 2 f 2 + ω 3 f 3 is the frame angular velocity, the component ω 1 does not change sign, and its arc length integral has the smallest value consistent with the boundary conditions. We consider construction of curves with rational minimal twist frames, based on the Pythagorean-hodograph curves of degree 7 that have rational rotation-minimizing Euler–Rodrigues frames ( e 1 ( ξ ) , e 2 ( ξ ) , e 3 ( ξ ) ) — i.e., the normal-plane vectors e 2 ( ξ ) , e 3 ( ξ ) have no rotation about the tangent e 1 ( ξ ) . A set of equations that govern the construction of such curves with prescribed initial/final points and tangents, and total arc length, is derived. For the resulting curves f 2 ( ξ ) , f 3 ( ξ ) are then obtained from e 2 ( ξ ) , e 3 ( ξ ) by a monotone rational normal-plane rotation, subject to the boundary conditions. A selection of computed examples is included to illustrate the construction, and it is shown that the desirable feature of a uniform rotation rate (i.e., ω 1 = constant ) can be accurately approximated.

A Novel Calibration Method Based on Vanishing Points and Lines for Linear Structured Light Vision Sensor

To meet the requirements of our own laboratory, the calibration method of linear structured light vision sensor based on the vanishing points and the vanihing lines is proposed. The concentric circles with a set of orthogonal diameters and known distance are designed as targets to calibrate the camera and linear structured light projector. Firstly, the orthogonal characteristics of the two vanishing points on orthogonal diameters are used to calibrate the internal and external parameters of the camera. Then the relationship between the normal vector of plane and the parameters in camera is applied to calibrate the normal vector of light plane. Finally, the forth parameter of light plane is obtained by using the coordinates of the known points in the target coordinate system. The experimental results show that the measurement average relative error of the sensor is 0.17%. It is proved that the linear structured light vision sensor with high accuracy and this is a feasible proposed calibration method.

C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

The problem of constructing a rational adapted frame (f1(ξ),f2(ξ),f3(ξ)) that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagorean-hodograph (PH) curve r(ξ) is addressed. PH curves are the only polynomial space curves that admit rational adapted frames, and the Euler–Rodrigues frame (ERF) is a fundamental instance of such frames. The ERF can be transformed into other rational adapted frame by applying a rationally-parametrized rotation to the normal-plane vectors. When orientation and angular velocity data at curve end points are given, a Hermite frame interpolant can be constructed using a complex quadratic polynomial that parametrizes the normal-plane rotation, by an extension of the method recently introduced to construct a rational minimal twist frame (MTF). To construct a rational adapted spline frame, a representation that resolves potential ambiguities in the orientation data is introduced. Based on this representation, a C1 rational adapted spline frame is constructed through local Hermite interpolation on each segment, using angular velocities estimated from a cubic spline that interpolates the frame phase angle relative to the ERF. To construct a C2 rational adapted spline frame, which ensures continuity of the angular acceleration, a complex-valued cubic spline is used to directly interpolate the complex exponentials of the phase angles at the nodal points.

A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

A rotation-minimizing frame $({\bf f}_1,{\bf f}_2,{\bf f}_3)$ on a space curve ${\bf r}(\xi)$ defines an orthonormal basis for $\mathbb{R}^3$ in which ${\bf f}_1={\bf r}'/|{\bf r}'|$ is the curve tangent, and the normal-plane vectors ${\bf f}_2$, ${\bf f}_3$ exhibit no instantaneous rotation about ${\bf f}_1$. Polynomial curves that admit rational rotation-minimizing frames (or RRMF curves) form a subset of the Pythagorean-hodograph (PH) curves, specified by integrating the form ${\bf r}'(\xi)={\cal A}(\xi)\,{\bf i}\,{\cal A}^*(\xi)$ for some quaternion polynomial ${\cal A}(\xi)$. By introducing the notion of rotation indicatrix and of core of the quaternion polynomial ${\cal A}(\xi)$, a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.

Geometry of the ringed surfaces in [formula omitted] that generate spatial Pythagorean hodographs

A Pythagorean-hodograph (PH) curver(t)=(x(t),y(t),z(t)) has the distinctive property that the components of its derivative r′(t) satisfy x′2(t)+y′2(t)+z′2(t)=σ2(t) for some polynomial σ(t). Consequently, the PH curves admit many exact computations that otherwise require approximations. The Pythagorean structure is achieved by specifying x′(t),y′(t),z′(t) in terms of polynomials u(t),v(t),p(t),q(t) through a construct that can be interpreted as a mapping from R4 to R3 defined by a quaternion product or the Hopf map. Under this map, r′(t) is the image of a ringed surface S(t,ϕ) in R4, whose geometrical properties are investigated herein. The generation of S(t,ϕ) through a family of four-dimensional rotations of a “base curve” is described, and the first fundamental form, Gaussian curvature, total area, and total curvature of S(t,ϕ) are derived. Furthermore, if r′(t) is non-degenerate, S(t,ϕ) is not developable (a non-trivial fact in R4). It is also shown that the pre-images of spatial PH curves equipped with a rotation-minimizing orthonormal frame (comprising the tangent and normal-plane vectors with no instantaneous rotation about the tangent) are geodesics on the surface S(t,ϕ). Finally, a geometrical interpretation of the algebraic condition characterizing the simplest non-trivial instances of rational rotation-minimizing frames on polynomial space curves is derived.

Understanding the Pincer - The Importance of Reference Plane Orientation on Acetabular Rim Evaluation

Objectives:Femoroacetabular impingement (FAI) is a common cause of young adult hip pain. FAI can result from an acetabular-sided bony lesion, or “pincer” lesion. A pincer lesion is defined as an area of linear contact between the acetabular rim and the femoral head-neck junction due to general or focal acetabular overcoverage. Three dimensional (3D) analysis of the acetabular rim morphology is essential to understand etiology and aciculate diagnosis of the of the pincer type FAI. A few studies have measured 3D geometry of the acetabular rim; in which the acetabular rim is described as a deviation from a reference plane. Therefore, the definition of the reference plane is critical to determine the acetabular rim geometry. The purpose of this study was to use 3D Computed Tomography (CT) modeling to evaluate the impact of varied acetabular orientation reference planes on the interpretation of acetabular rim abnormalities, with the goal to determine the ideal reference plane for future study use.Methods:3D CT modeling was performed on five hip joints of patients who underwent hip arthroscopy with acetabular trimming for a presumed pincer lesion. These models were exported into point-cloud models. An acetabular 3D model was automatically created within 10 mm from the femoral head surface (Fig. 1A, B). The acetabular articular surface and rim were separated with a threshold of 5 mm, which provided an acetabular rim model with a band width of 5 mm (Fig. 1C). A local coordinate system was defined with the acetabular notch midpoint being 6 o’clock. A best-fit plane of the acetabular rim was determined by the least square method using two different acetabular rim models: 1) a model excluding the acetabular notch (plane A) and 2) a model excluding the acetabular notch and superior region from 10:30 to 3:00 (3:00 being anterior) (plane B). The acetabular rim model was transformed into a cylindrical coordinate system with an axis determined by a normal vector of the plane. The final acetabular rim model consisting of 120 points with 3° increments was created by searching the outermost points of the rim. The acetabular center was determined using best-fit sphere of the articular surface model. A reference plane including the center point was determined with orientations determined by normal vectors of the acetabular rim planes (Fig. 1C, yellow line). The 3D geometry of the rim was described by subtended angles from the normal vector of the reference plane.Results:Three distinct peaks were noted at anteroinferior (AI), anterosuperior (AP) and posteroinferior (PI) regions (Fig. 2 A, B). While the AI and PI peaks measured with the plane A were higher than that measured with plane B, the AP peak measured with plane A was lower than that measured with plane B (Fig. 2C). The angle between the normal vectors of plane A and B was 13.7±3.5°.Conclusion:The findings demonstrate that the orientation of the reference plane is critical to the 3D measurement of the acetabular rim. Since bony prominence in the anterosuperior region has been considered as the pathogenesis of the impingement, the reference plane including this region may cause underestimation of the bony lesion. An appropriate determination of the reference plane is crucial for evaluation of the bony lesion in the pincer FAI patient.

손동작을 이용한 상호작용 증강현실 시스템

본 논문에서는 컴퓨터 비전 기술 기반으로 사용자의 손 움직임과 가상객체 사이의 자연스러운 상호작용을 위한 증강현실 시스템을 제안한다. 기존의 증강현실 시스템은 마커를 이용하거나 트랙커 같은 센서를 직접 조작해야 했기 때문에 사용자에게 불편함을 제공한다. 우리는 이러한 문제점을 해결하기 위해서 손 움직임을 증강현실 시스템에 적용한다. 또한 가상객체의 움직임에 물리적 현상을 적용하여 현실감을 부여한다. 제안한 시스템은 마커를 이용하여 기하학적 정보를 얻어 가상의 공이 움직이는 가상공간, 벽돌 등의 시스템 환경을 구성하였으며, 사용자의 손 움직임은 테이핑을 통하여 특징점을 추출하여 정보를 얻는다. 이 특징점을 이용하여 손 위에 가상 평면을 안정적으로 정합한다. 가상의 공의 움직임은 포물선 방정식을 이용하여 기본적으로 포물선 움직임을 보이고, 평면 또는 벽돌에 충돌이 발생할 경우에는 가상 공의 위치와 평면의 법선벡터를 이용하여 가상 공의 움직임을 보였다. 실험을 통해 충돌 시 발생되는 잘못된 공의 위치에 대해 보정되는 과정을 보였고, 증강된 가상객체의 떨림과 수행속도에 대해 마커를 이용한 시스템과 비교하여 제안한 시스템으로 대체할 수 있다는 것을 확인하였다. In this paper, We propose Augmented Reality (AR) System for the interaction between user's hand motion and virtual object motion based on computer vision. The previous AR system provides inconvenience to user because the users have to control the marker and the sensor like a tracker. We solved the problem through hand motion and provide the convenience to the user. Also the motion of virtual object using a physical phenomenon gives a reality. The proposed system obtains geometrical information by the marker and hand. The system environments like virtual space of moving virtual ball and bricks are made by using the geometrical information and user's hand motion is obtained from the hand's information with extracted feature point through the taping hand. And it registers a virtual plane stably by getting movement of the feature points. The movement of the virtual ball basically is parabolic motion with a parabolic equation. When the collision occurs either the planes or the bricks, we show movement of the virtual ball with ball position and normal vector of plane and the ball position is faulted. So we showed corrected ball position through experiment. and we proved that this system can replaced the marker system to compare to jitter of augmented virtual object and progress speed with it.

A complete classification of quintic space curves with rational rotation-minimizing frames

An adapted orthonormal frame ( f 1 , f 2 , f 3 ) on a space curve r ( t ) , where f 1 = r ′ / | r ′ | is the curve tangent, is rotation-minimizing if its angular velocity satisfies ω ⋅ f 1 ≡ 0 , i.e., the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . The simplest space curves with rational rotation-minimizing frames (RRMF curves) form a subset of the quintic spatial Pythagorean-hodograph (PH) curves, identified by certain non-linear constraints on the curve coefficients. Such curves are useful in motion planning, swept surface constructions, computer animation, robotics, and related fields. The condition that identifies the RRMF quintics as a subset of the spatial PH quintics requires a rational expression in four quadratic polynomials u ( t ) , v ( t ) , p ( t ) , q ( t ) and their derivatives to be reducible to an analogous expression in just two polynomials a ( t ) , b ( t ) . This condition has been analyzed, thus far, in the case where a ( t ) , b ( t ) are also quadratic, the corresponding solutions being called Class I RRMF quintics. The present study extends these prior results to provide a complete categorization of all possible PH quintic solutions to the RRMF condition. A family of Class II RRMF quintics is thereby newly identified, that correspond to the case where a ( t ) , b ( t ) are linear. Modulo scaling/rotation transformations, Class II curves have five degrees of freedom, as with the Class I curves. Although Class II curves have rational RMFs that are only of degree 6–as compared to degree 8 for Class I curves–their algebraic characterization is more involved than for the latter. Computed examples are used to illustrate the construction and properties of this new class of RRMF quintics. A novel approach for generating RRMF quintics, based on the sum-of-four-squares decomposition of positive real polynomials, is also introduced and briefly discussed.

Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves

A rotation-minimizing frame on a space curve r ( t ) is an orthonormal basis ( f 1 , f 2 , f 3 ) for R 3 , where f 1 = r ′ / | r ′ | is the curve tangent, and the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . Such frames are useful in spatial path planning, swept surface design, computer animation, robotics, and related applications. The simplest curves that have rational rotation-minimizing frames (RRMF curves) comprise a subset of the quintic Pythagorean-hodograph (PH) curves, and two quite different characterizations of them are currently known: (a) through constraints on the PH curve coefficients; and (b) through a certain polynomial divisibility condition. Although (a) is better suited to the formulation of constructive algorithms, (b) has the advantage of remaining valid for curves of any degree. A proof of the equivalence of these two different criteria is presented for PH quintics, together with comments on the generalization to higher-order curves. Although (a) and (b) are both sufficient and necessary criteria for a PH quintic to be an RRMF curve, the (non-obvious) proof presented here helps to clarify the subtle relationships between them.

Rational rotation-minimizing frames on polynomial space curves of arbitrary degree

A rotation-minimizing adapted frame on a space curve r ( t ) is an orthonormal basis ( f 1 , f 2 , f 3 ) for R 3 such that f 1 is coincident with the curve tangent t = r ′ / | r ′ | at each point and the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . Such frames are of interest in applications such as spatial path planning, computer animation, robotics, and swept surface constructions. Polynomial curves with rational rotation-minimizing frames (RRMF curves) are necessarily Pythagorean-hodograph (PH) curves–since only the PH curves possess rational unit tangents–and they may be characterized by the fact that a rational expression in the four polynomials u ( t ) , v ( t ) , p ( t ) , q ( t ) that define the quaternion or Hopf map form of spatial PH curves can be written in terms of just two polynomials a ( t ) , b ( t ) . As a generalization of prior characterizations for RRMF cubics and quintics, a sufficient and necessary condition for a spatial PH curve of arbitrary degree to be an RRMF curve is derived herein for the generic case satisfying u 2 ( t ) + v 2 ( t ) + p 2 ( t ) + q 2 ( t ) = a 2 ( t ) + b 2 ( t ) . This RRMF condition amounts to a divisibility property for certain polynomials defined in terms of u ( t ) , v ( t ) , p ( t ) , q ( t ) and their derivatives.