At present, more and more young people join the basketball game. The spot-up shot is the foundation of modern basketball technology. Basketball shooting percentage is directly related to the initial speed and angle of the shot. We discussed the best shot angle from different angles [1]. The best shooting angle has always been a research hotspot in basketball theory. At present, there are many ideal basketball trajectory models in academia. Some scholars have used the variational method to study the problem of the best shooting angle. They discussed the shooting angle in detail from three angles: the best angle against angular deviation, the best angle against speed error, and the most effort-saving shot. They produce accurate quantitative results that are realistic. According to the ideal shooting trajectory model, some scholars have studied in detail the best position for the highest point of the shot. The author further gives the double 1/3 method of basketball shooting training under visual observation and improves the shooting percentage.

The no-resistance shooting trajectory model can analyze the best shot angle with a complex variational method. But the algorithm principle is complex, and the process is cumbersome. And its geometric meaning is the angle bisector. Therefore, this method is not conducive to guiding the actual shooting. The best release angle can also be estimated based on the best position for the estimated shot peak [2]. However, if the best shot angle is not directly given, it is also not conducive to guiding the actual shooting. Parabolic trajectory model with drag Although the functional equation for the best throw angle of the shot put can be derived from the principle of parabola, it is not the best throw angle for basketball. The process and result are quite complicated.

This paper establishes a model of shooting trajectory without resistance. We use the discriminant of the quadratic equation to derive the best shot angle simply and clearly. Its geometric meaning is a straight line between two points [3]. This helps guide the actual shot. We expand the model of shot trajectories with resistance based on Taylor. We perfectly unify the model of the shot trajectory with resistance and the model of the shot trajectory without resistance.

Suppose the basketball is a particle and ignore air resistance. At this point, we can get the basketball equation and curve as:

Where

We first fix

Then we get the best angle against angular deviation as follows:

But air resistance is unavoidable [4]. Assume that air resistance

We can directly eliminate

This formula is too complicated and is not conducive to analyzing properties such as the optimum angle against angular deviation. Some scholars have simplified and approximated the above formula to find the shot angle when the shot put is thrown farthest. We expand ^{−kt/m} using series. The formula (5) is simplified to:

This is the same relationship between

We add some terms without considering air resistance when there is air resistance. If the expansion is represented by an infinite series, the third term in equation (9) contains the resistance coefficient

The model of the shot trajectory ignoring resistance is expressed as follows:

Among them,

Eliminate time parameter

We regard tan

So

That is, the minimum square of the initial velocity is:

We can find out by substituting the initial velocity at this time into equation (13):

This angle is the most labor-saving release angle [8]. The geometric meaning of this angle is as follows. As shown in Figure 1,

It is demonstrated below that this angle is completely consistent with the angle

The geometric meaning in the literature is shown in Figure 2. Know
^{2}

The final result of the two methods is the same. In the literature, the complex variational method is used to obtain it. But this paper only uses the discriminant of the simple quadratic equation. In addition, the geometric meaning in the literature is the angle bisector and is not conducive to guiding the actual shooting. In this paper, the geometric meaning is a straight line. This method facilitates the actual shot aiming.

The trajectory simulation equation with resistance is as follows

This formula is very complicated, and it is difficult to directly find the most labor-saving angle of the shot [9]. Therefore, this formula is approximately simplified in this paper. We substitute the most effortless release angle and release speed into

Assuming
^{4} − 8^{2} − 4^{2} ^{2} ^{2} ≥ 0,

The minimum square of the initial velocity is:

We can obtain

Although athletes training shots on the basketball court can increase the feel of the shot, they cannot see the trajectory of the basketball movement [11]. This paper develops an augmented reality-based shooting trajectory simulation system. This allows players to see the trajectory of their shots in real-time. The system software structure diagram is shown in Figure 4, and the system working diagram is shown in Figure 5.

The main purpose of the front module of the system is to obtain the shooting speed and shooting angle. The way to obtain it is to record the hand's position in real-time according to the magnetic tracker and the data glove [12]. The trajectory curve is fitted according to a certain algorithm. We end up with release speed and release angle.

The modules in the system are first based on the shot speed and shot angle obtained before the system. We render the trajectory of virtual basketball according to the previous illustrative model. Then it is judged whether the virtual basketball falls into the virtual basket.

The main purpose of the rear module of the system is based on the virtual basketball trajectory in front and whether it falls into the virtual basket. We give instructive advice to athletes.

The shooting trajectory can be approximated as a parabola. This paper first gives the most labor-saving angle of shot through theoretical analysis. The complex shooting trajectory is simplified by Taylor expansion when there is resistance. The algorithm proves that when the air resistance tends to 0, it is the same as the shooting trajectory without considering the air resistance. Secondly, through the discriminant of quadratic equation, the acquisition of the most labor-saving shooting angle without considering air resistance is given. We give its geometric meaning. The geometric meaning of the shooting angle given in this article is a straight line between two points. This is great for coaching the actual shot. Finally, this paper designs an augmented reality system. The system allows players to see the trajectory of the virtual basketball and the virtual basket in real-time. This effectively instructs the athlete to change his shooting speed and shooting angle in real-time.

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