Calculating Connections Using Coordinate Objects (coord)

1. Introduction

In differential geometry and physics, connections are fundamental tools for describing how vector fields are "parallel transported" in curved spaces. Traditional approaches rely on abstract Christoffel symbols and tensor operations, whereas this article introduces an intuitive computational method based on coordinate objects (coord), which is well-suited for computer implementations (e.g., robotics, computer graphics).

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2. Definition of Coordinate Object (coord)

We define a three-dimensional coordinate object coord with the following members:

struct coord {
    vec3 ux, uy, uz; // Three orthogonal basis vectors (local coordinate directions)
    vec3 s;          // Scaling factors
    vec3 o;          // Origin position
};

• Construction Methods:
  • From basis vectors: coord(vec3 ux, vec3 uy, vec3 uz)
  • From Euler angles: coord(float pitch, float yaw, float roll)

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3. Local Implementation of Connection: Gradient Coordinate System Gu

The core idea of a connection is to describe transformations between neighboring coordinate systems. We achieve this using the gradient coordinate system Gu:

// Compute the gradient coordinate system between two neighboring coordinates C1 and C2
coord Gu = C1 / C2 - I; // I is the identity coordinate system
coord G_intrinsic = C2 / C1 / c2 - I / c1; // Intrinsic connection (embedded surfaces)

• Physical Interpretation:
  Gu is the linear approximation of the transformation from C1 to C2, analogous to the connection coefficients Γ^k_{ij} in differential geometry.

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4. Vector Transport and Covariant Derivative

(1) Vector Transport

Transport a vector V1 from coordinate system C1 to C2:

vec3 V2 = V1 * (C2 / C1);

• Geometric Interpretation:
  If C1 and C2 are local coordinate systems at neighboring points on a manifold, V2 is the result of "parallel transporting" V1.

(2) Covariant Derivative

The rate of change (covariant derivative ∇_u V) of a vector field V along direction u can be approximated by finite differences:

coord Cu = ...; // Differential coordinate system in direction u
vec3 deltaV = V * Gu;       // Gu = C1 / C2 - I
vec3 covDerivative = deltaV / Cu; // Covariant derivative

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5. Curvature Calculation

Curvature reflects the path-dependence of parallel transport. It is computed via the non-commutativity of gradient coordinate systems:

coord Guv = Gu * Gv - Gv * Gu; // [Gu, Gv] Lie bracket

• Relation to Traditional Curvature Tensors:
  Guv corresponds to the local representation of the Riemann curvature tensor R(X,Y)Z.

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6. Code Implementation Examples

(1) Coordinate System Multiplication and Division

// Transform a vector from local coordinate system C to world coordinates
vec3 operator*(const vec3& p, const coord& c) {
    return c.ux * (c.s.x * p.x) + c.uy * (c.s.y * p.y) + c.uz * (c.s.z * p.z) + c.o;
}

// Composite transformation of coordinate systems (analogous to Lie group multiplication)
coord operator*(const coord& c1, const coord& c2) {
    coord rc;
    rc.ux = c1.ux.x * c2.ux + c1.ux.y * c2.uy + c1.ux.z * c2.uz;
    rc.s = c1.s * c2.s;
    rc.o = c1.o.x * c2.ux + c1.o.y * c2.uy + c1.o.z * c2.uz + c2.o;
    return rc;
}

(2) Connection Calculation

// Compute the gradient coordinate system Gu
coord computeGu(const coord& C1, const coord& C2) {
    return C1 / C2 - identity_coord(); // identity_coord() is the identity coordinate system
}

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7. Application Scenarios

1. Robotics Kinematics:
   Compute the transfer of robotic arm joint velocities across different coordinate systems (avoiding Euler angle singularities).
2. Computer Graphics:
   Implement texture mapping or lighting calculations on curved surfaces.
3. Physics Simulations:
   Model spacetime curvature effects in general relativity.

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8. Conclusion

• Coordinate object coord transforms abstract connection operations into intuitive multiplication and division.
• Gradient coordinate system Gu is a local implementation of connections, used for vector transport and curvature calculations.
• Advantages:
  • Avoids complex tensor notation.
  • Can be directly embedded in C++/Python code, suitable for numerical computations.

This approach makes the theory of connections in differential geometry programmable and accessible!