Constant envelope (CE) precoding can efficiently control the peak-to-average power ratio (PAPR) and improve the power efficiency of power amplifiers in large-scale antenna array systems. Antenna subset selection (ASS), combined with CE precoding, can further improve power efficiency by using a part of antennas to combine the desired signal. However, due to the inherent nonlinearity, the joint optimization of CE precoding and ASS is very challenging and satisfactory solutions are yet not available. In this paper, we present new methods for CE precoding and ASS optimization from a geometric perspective. First, we show the equivalence between the CE precoder design and a polygon construction problem in the complex plane, thus transforming the algebraic problem into a geometric problem. Aiming to minimize the computational complexity, we further transform the CE precoder design into a triangle construction problem, and propose a novel algorithm to achieve the optimal CE precoder with only linear complexity in the number of used antennas. Then, we investigate the joint optimization of ASS and CE precoding to minimize the total transmit power while satisfying the QoS requirement. Based on the geometric interpretation, we develop an efficient ASS algorithm, which, using only addition and comparison operations, is guaranteed to find the globally optimal solution and provides robustness to channel uncertainty. The complexity of the proposed ASS algorithm is at most quadratic in the number of antennas in the worst case. The optimality and superiority of the proposed geometric methods are demonstrated via numerical results.