# A Parametric Identification Method of Human Gait Differences and its Application in Rehabilitation
## 2 Methods
### Mathematical Model Application
#### 2.3.2 Geometric Properties  
The local shapes of the GE-Bezier curve can be adjusted by changing the shape parameters without modifying the control point. The curve will gradually approach the corresponding control point with the increase of the shape parameter value. On the contrary, it will gradually move away, which means that a family of curves with different detail shapes can be generated by Equation (1) with the alteration of the shape parameters after the control points are determined.  
For instance, assuming that the whole shape of a curve can be defined by five control points (\(P_{1}\),..., \(P_{5}\)), then three adjustable shape parameters and their value ranges can be calculated through Equations (1) and (2). Figure 1 depicts how these three shape parameters influenced the local shapes of the curve. Figure 2a shows the overall shape of the curve (in solid line) when all shape parameters were median values, i.e., (\(\lambda_{0},\,\lambda_{1},\,\lambda_{2}\)) = (\(-1,\,1.5,\,-1\)). When the first shape parameter was the maximum and the minimum of the range of values, and the other two parameters stayed at the median, i.e., (\(\lambda_{0},\,\lambda_{1},\,\lambda_{2}\)) = (\(1,\,1.5,\,-1\)) and (\(\lambda_{0},\,\lambda_{1},\,\lambda_{2}\)) = (\(-3,\,1.5,\,-1\)), then, in Figure 2b, the first wave of the curve displayed the highest (in the dash-dotted line), and the lowest (in dashed line), respectively, while the other two waves changed little. In Figure 2c, when the maximum and minimum values were given to the second parameter and the others were intermediate values, for (\(\lambda_{0},\,\lambda_{1},\,\lambda_{2}\)) = (\(-1,\,3,\,-1\)) in dash-dotted line, for (\(\lambda_{0},\,\lambda_{1},\,\lambda_{2}\)) = (\(-1,\,0,\,-1\)) in the dashed line, the second wave of the curve appeared from approachingthe control point, \(P_{3}\), to far away from it, and the others remained stable. Similarly, in Figure 2d, only the third parameter was changed, with (\(\lambda_{0}\), \(\lambda_{1}\), \(\lambda_{2}\)) = (\(-1\), \(1.5\), \(1\)) in the dash-dotted line, and (\(\lambda_{0}\), \(\lambda_{1}\), \(\lambda_{2}\)) = (\(-1\), \(1.5\), \(-3\)) in the dashed line. Then, the third wave of the curve went from high to low. In order to acquire the total range of shape adjustment for the curve within the value range of parameters, let (\(\lambda_{0}\), \(\lambda_{1}\), \(\lambda_{2}\)) = (\(1\), \(0\), \(1\)) in the dash-dotted line and (\(\lambda_{0}\), \(\lambda_{1}\), \(\lambda_{2}\)) = (\(-3\), \(-3\)) in the dashed line. The grey area in Figure 2e is the whole range of curve shape adjustment under these five control points.  
Therefore, (1) in order to ensure that the local shape of the curve has sufficient adjustment ranges when approaching and moving away from corresponding control points, the average curve (with median values of shape parameters) should be chosen to be fitted by Equation (1) to determine the control points' numbers and coordinates. In this article, it refers to the three average angle curves of joints (hip, knee, and ankle) from the normal group to be fitted. (2) The control points of the normal group are taken as the reference commonality points for other participants, and the ranges of the shape parameters of the normal group are also taken as the normal ranges of the normal gait. In this way, if the shape parameters of other participants belong to the normal ranges under the reference control points, the gait is normal; otherwise, it indicates abnormal gait.