Kalman Filter and Orthogonal Projections: A Geometric Visualization

① This example visualizes a simple Kalman filter through vector projections, turning the iterative filtering algorithm into intuitive geometric operations in space. ② Core mathematical concepts include conditional probability, random variables, variance, covariance, linear regression, orthogonal projection, and the variance update equations of the Kalman filter, highlighting the deep link between geometry and probability/statistics. ③ Construction details: Using GeoGebra commands such as Intersect, Vector, Plane, and OrthogonalLine, the construction involves objects B, A, x0, C, v1, z1, x1est, B1, D, v2, z2, E, F, a, f, Z1, z2ort, x2est, B2, x1err, x2err, b, and z2err, which progressively illustrate the geometric meaning of state estimation and error via vector decomposition and orthogonal projection. ④ Suitable for university-level courses in probability, statistics, stochastic processes, or signal processing, and for graduate students seeking the geometric essence of Kalman filtering; also useful for instructors creating interactive demonstrations.