Piecewise Linear Approximation (PLA) is a widely used technique for compressing time series by approximating data points with line segments, reducing storage and transmission costs. In this paper, we propose a novel vertical distance metric as a replacement for the conventional Y-axis distance in the max-error PLA framework, aiming to improve compression efficiency while preserving approximation accuracy. We further propose two optimal linear-time PLA algorithms using vertical distance, designed to generate the minimum number of disconnected and connected segments, respectively. Extended experimental results demonstrate that our methods outperform traditional approaches in terms of compression effectiveness, achieving higher compression ratios. For specific datasets, our methods significantly outperform traditional techniques in terms of compression ratios.